#set document(title: "3.3 Roots and Radicals", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 3.3#h(0.6em)Roots and Radicals In Integer Exponents we saw that inverse variation can be expressed as a power function by using negative exponents. We can also use exponents to denote square roots and other radicals. === #math.equation(block: false, alt: "n")[$n$]th Roots Recall that #math.equation(block: false, alt: "s")[$s$] is a square root of #math.equation(block: false, alt: "b")[$b$] if #math.equation(block: false, alt: "s squared equals b")[$s^(2) = b$], and #math.equation(block: false, alt: "s")[$s$] is a cube root of #math.equation(block: false, alt: "b")[$b$] if #math.equation(block: false, alt: "s cubed equals b")[$s^(3) = b$]. In a similar way, we can define the fourth, fifth, or sixth root of a number. For instance, the fourth root of #math.equation(block: false, alt: "b")[$b$] is a number #math.equation(block: false, alt: "s")[$s$] whose fourth power is #math.equation(block: false, alt: "b")[$b$]. In general, we make the following definition. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[#math.equation(block: false, alt: "n")[$n$]th Roots] #math.equation(block: false, alt: "s")[$s$] is called an #strong[#math.equation(block: false, alt: "n")[$n$]th root of #math.equation(block: false, alt: "b")[$b$]] if #math.equation(block: false, alt: "s to the power n equals b")[$s^(n) = b$]. ] We use the symbol #math.equation(block: false, alt: "the nth root of b")[$root(n, b)$] to denote the #math.equation(block: false, alt: "n")[$n$]th root of #math.equation(block: false, alt: "b")[$b$]. An expression of the form #math.equation(block: false, alt: "the nth root of b")[$root(n, b)$] is called a #strong[radical], #math.equation(block: false, alt: "b")[$b$] is called the #strong[radicand], and #math.equation(block: false, alt: "n")[$n$] is called the #strong[index of the radical]. #examplebox("Example 1")[][ + #math.equation(block: false, alt: "the 4th root of 81 equals 3")[$root(4, 81) = 3$] because #math.equation(block: false, alt: "3 to the power 4 equals 81")[$3^(4) = 81$] + #math.equation(block: false, alt: "the 5th root of 32 equals 2")[$root(5, 32) = 2$] because #math.equation(block: false, alt: "2 to the power 5 equals 32")[$2^(5) = 32$] + #math.equation(block: false, alt: "the 6th root of 64 equals 2")[$root(6, 64) = 2$] because #math.equation(block: false, alt: "2 to the power 6 equals 64")[$2^(6) = 64$] + #math.equation(block: false, alt: "the 4th root of 1 equals 1")[$root(4, 1) = 1$] because #math.equation(block: false, alt: "1 to the power 4 equals 1")[$1^(4) = 1$] + #math.equation(block: false, alt: "the 5th root of 100 , 000 equals 10")[$root(5, 100 "," 000) = 10$] because #math.equation(block: false, alt: "10 to the power 5 equals 100 , 000")[$10^(5) = 100 , 000$] ] Evaluate each radical. + #math.equation(block: false, alt: "the 4th root of 16 equals")[$root(4, 16) =$]\_\_\_\_\_ + #math.equation(block: false, alt: "the 5th root of 243 equals")[$root(5, 243) =$]\_\_\_\_\_ #solutionbox[ + #math.equation(block: false, alt: "2")[$2$] + #math.equation(block: false, alt: "3")[$3$] ] Evaluate each radical. + #math.equation(block: false, alt: "the 4th root of 16")[$root(4, 16)$] + #math.equation(block: false, alt: "the 5th root of 243")[$root(5, 243)$] #solutionbox[ + #math.equation(block: false, alt: "2")[$2$] + #math.equation(block: false, alt: "3")[$3$] ] === Exponential Notation for Radicals A convenient notation for radicals uses fractional exponents. Consider the expression #math.equation(block: false, alt: "9 to the power 1 / 2")[$9^(1 / 2)$]. What meaning can we attach to an exponent that is a fraction? The third law of exponents says that when we raise a power to a power, we multiply the exponents together: #math.equation(block: true, alt: "open parenthesis x to the power a close parenthesis to the power b equals x to the power a b")[$attach(( x^(a) ), t: b) = x^(a b)$] Therefore, if we square the number #math.equation(block: false, alt: "9 to the power 1 / 2")[$9^(1 / 2)$], we get #math.equation(block: true, alt: "open parenthesis 9 to the power 1 / 2 close parenthesis squared equals 9 to the power open parenthesis 1 / 2 close parenthesis open parenthesis 2 close parenthesis equals 9 to the power 1 equals 9")[$attach(( 9^(1 / 2) ), t: 2) = 9^(( 1 / 2 ) ( 2 )) = 9^(1) = 9$] Thus, #math.equation(block: false, alt: "9 to the power 1 / 2")[$9^(1 / 2)$] is a number whose square is #math.equation(block: false, alt: "9")[$9$]. But this means that #math.equation(block: false, alt: "9 to the power 1 / 2")[$9^(1 / 2)$] is a square root of #math.equation(block: false, alt: "9")[$9$], or #math.equation(block: true, alt: "9 to the power 1 / 2 equals the square root of 9 equals 3")[$9^(1 / 2) = sqrt(9) = 3$] In general, any nonnegative number raised to the #math.equation(block: false, alt: "1 / 2")[$1 / 2$] power is equal to the positive square root of the number, or #math.equation(block: true, alt: "a to the power 1 / 2 equals the square root of a")[$a^(1 / 2) = sqrt(a)$] #examplebox("Example 2")[][ + #math.equation(block: false, alt: "25 to the power 1 / 2 equals 5")[$25^(1 / 2) = 5$] + #math.equation(block: false, alt: "minus 25 to the power 1 / 2 equals minus 5")[$− 25^(1 / 2) = − 5$] + #math.equation(block: false, alt: "open parenthesis minus 25 close parenthesis to the power 1 / 2")[$( − 25 )^(1 / 2)$] is not a real number. + #math.equation(block: false, alt: "0 to the power 1 / 2 equals 0")[$0^(1 / 2) = 0$] ] Evaluate each power. + #math.equation(block: false, alt: "4 to the power 1 / 2 equals")[$4^(1 / 2) =$]\_\_\_\_\_ + #math.equation(block: false, alt: "4 to the power minus 2 equals")[$4^(− 2) =$]\_\_\_\_\_ + #math.equation(block: false, alt: "4 to the power minus 1 / 2 equals")[$4^(− 1 / 2) =$]\_\_\_\_\_ + #math.equation(block: false, alt: "open parenthesis the fraction 1 over 4 close parenthesis to the power 1 / 2 equals")[$attach(( display(frac(1, 4)) ), t: 1 / 2) =$]\_\_\_\_\_ #solutionbox[ + #math.equation(block: false, alt: "2")[$2$] + #math.equation(block: false, alt: "the fraction 1 over 16")[$display(frac(1, 16))$] + #math.equation(block: false, alt: "the fraction 1 over 2")[$display(frac(1, 2))$] + #math.equation(block: false, alt: "the fraction 1 over 2")[$display(frac(1, 2))$] ] Evaluate each power. + #math.equation(block: false, alt: "4 to the power 1 / 2")[$4^(1 / 2)$] + #math.equation(block: false, alt: "4 to the power minus 2")[$4^(− 2)$] + #math.equation(block: false, alt: "4 to the power minus 1 / 2")[$4^(− 1 / 2)$] + #math.equation(block: false, alt: "open parenthesis the fraction 1 over 4 close parenthesis to the power 1 / 2")[$attach(( display(frac(1, 4)) ), t: 1 / 2)$] #solutionbox[ + #math.equation(block: false, alt: "2")[$2$] + #math.equation(block: false, alt: "the fraction 1 over 16")[$display(frac(1, 16))$] + #math.equation(block: false, alt: "the fraction 1 over 2")[$display(frac(1, 2))$] + #math.equation(block: false, alt: "the fraction 1 over 2")[$display(frac(1, 2))$] ] The same reasoning works for roots with any index. For instance, #math.equation(block: false, alt: "8 to the power 1 / 3")[$8^(1 / 3)$] is the cube root of #math.equation(block: false, alt: "8")[$8$], because #math.equation(block: true, alt: "open parenthesis 8 to the power 1 / 3 close parenthesis cubed equals 8 to the power open parenthesis 1 / 3 close parenthesis open parenthesis 3 close parenthesis equals 8 to the power 1 equals 8")[$attach(( 8^(1 / 3) ), t: 3) = 8^(( 1 / 3 ) ( 3 )) = 8^(1) = 8$] In general, we make the following definition for fractional exponents. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Exponential Notation for Radicals] For any integer #math.equation(block: false, alt: "n greater than or equal to 2")[$n ≥ 2$] and for #math.equation(block: false, alt: "a greater than or equal to 0")[$a ≥ 0$], #math.equation(block: true, alt: "a to the power 1 / n equals the nth root of a")[$a^(1 / n) = root(n, a)$] ] #examplebox("Example 3")[][ + #math.equation(block: false, alt: "81 to the power 1 / 4 equals the 4th root of 81 equals 3")[$81^(1 / 4) = root(4, 81) = 3$] + #math.equation(block: false, alt: "125 to the power 1 / 3 equals the cube root of 125 equals 5")[$125^(1 / 3) = root(3, 125) = 5$] ] #notebox("Caution", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Note that #math.equation(block: true, alt: "25 to the power 1 / 2 not equal to the fraction 1 over 2 open parenthesis 25 close parenthesis and 125 to the power 1 / 3 not equal to the fraction 1 over 3 open parenthesis 125 close parenthesis")[$25^(1 / 2) ≠ frac(1, 2) ( 25 ) " " " " " and " " " " " 125^(1 / 3) ≠ frac(1, 3) ( 125 )$] An exponent of #math.equation(block: false, alt: "the fraction 1 over 2")[$display(frac(1, 2))$] denotes the square root of its base, and an exponent of #math.equation(block: false, alt: "the fraction 1 over 3")[$display(frac(1, 3))$] denotes the cube root of its base. ] Write each power with radical notation, and then evaluate. + #math.equation(block: false, alt: "32 to the power 1 / 5")[$32^(1 / 5)$] is #linebreak() \_\_\_\_\_ #linebreak() #math.equation(block: false, alt: "32 to the power 1 / 5 equals")[$32^(1 / 5) =$]\_\_\_\_\_ + #math.equation(block: false, alt: "625 to the power 1 / 4")[$625^(1 / 4)$] is #linebreak() \_\_\_\_\_ #linebreak() #math.equation(block: false, alt: "625 to the power 1 / 4 equals")[$625^(1 / 4) =$]\_\_\_\_\_ #solutionbox[ + #math.equation(block: false, alt: "32 to the power 1 / 5 equals the 5th root of 32 equals 2")[$32^(1 / 5) = root(5, 32) = 2$] + #math.equation(block: false, alt: "625 to the power 1 / 4 equals the 4th root of 625 equals 5")[$625^(1 / 4) = root(4, 625) = 5$] ] Write each power with radical notation, and then evaluate. + #math.equation(block: false, alt: "32 to the power 1 / 5 equals bar equals bar")[$32^(1 / 5) = underline(#h(2em)) = underline(#h(2em))$] + #math.equation(block: false, alt: "625 to the power 1 / 4 equals bar equals bar")[$625^(1 / 4) = underline(#h(2em)) = underline(#h(2em))$] #solutionbox[ + #math.equation(block: false, alt: "32 to the power 1 / 5 equals the 5th root of 32 equals 2")[$32^(1 / 5) = root(5, 32) = 2$] + #math.equation(block: false, alt: "625 to the power 1 / 4 equals the 4th root of 625 equals 5")[$625^(1 / 4) = root(4, 625) = 5$] ] Of course, we can use decimal fractions for exponents as well. For example, #math.equation(block: true, alt: "the square root of a equals a to the power 1 / 2 equals a to the power 0.5 and the 4th root of a equals a to the power 1 / 4 equals a to the power 0.25")[$sqrt(a) = a^(1 / 2) = a^(0.5) " " " " " " " " " and " " " " " " " " " root(4, a) = a^(1 / 4) = a^(0.25)$] Which of the following expressions is not equivalent to the other three? \_\_\_\_\_ #solutionbox[ #math.equation(block: false, alt: "the fraction 1 over B to the power 8")[$display(frac(1, B^(8)))$] is not equivalent to the other three expressions. ] Which of the following expressions is not equivalent to the other three? + #math.equation(block: false, alt: "B to the power the fraction 1 over 8")[$B^(frac(1, 8))$] + #math.equation(block: false, alt: "the 8th root of B")[$root(8, B)$] + #math.equation(block: false, alt: "B to the power 0.125")[$B^(0.125)$] + #math.equation(block: false, alt: "the fraction 1 over B to the power 8")[$display(frac(1, B^(8)))$] #examplebox("Example 4")[][ + #math.equation(block: false, alt: "100 to the power 0.5 equals the square root of 100 equals 10")[$100^(0.5) = sqrt(100) = 10$] + #math.equation(block: false, alt: "16 to the power 0.25 equals the 4th root of 16 equals 2")[$16^(0.25) = root(4, 16) = 2$] ] Write each power with radical notation, and then evaluate. + #math.equation(block: false, alt: "100 , 000 to the power 0.2")[$100 , 000^(0.2)$] is #linebreak() \_\_\_\_\_ #linebreak() #math.equation(block: false, alt: "100 , 000 to the power 0.2 equals")[$100 , 000^(0.2) =$]\_\_\_\_\_ + #math.equation(block: false, alt: "81 to the power 0.25")[$81^(0.25)$] is #linebreak() \_\_\_\_\_ #linebreak() #math.equation(block: false, alt: "81 to the power 0.25 equals")[$81^(0.25) =$]\_\_\_\_\_ #solutionbox[ + #math.equation(block: false, alt: "100 , 000 to the power 0.2 equals the 5th root of 100 , 000 equals 10")[$100 , 000^(0.2) = root(5, 100 "," 000) = 10$] + #math.equation(block: false, alt: "81 to the power 0.25 equals the 4th root of 81 equals 3")[$81^(0.25) = root(4, 81) = 3$] ] Write each power with radical notation, and then evaluate. + #math.equation(block: false, alt: "100 , 000 to the power 0.2 equals bar equals bar")[$100 , 000^(0.2) = underline(#h(2em)) = underline(#h(2em))$] + #math.equation(block: false, alt: "81 to the power 0.25 equals bar equals bar")[$81^(0.25) = underline(#h(2em)) = underline(#h(2em))$] #solutionbox[ + #math.equation(block: false, alt: "100 , 000 to the power 0.2 equals the 5th root of 100 , 000 equals 10")[$100 , 000^(0.2) = root(5, 100 "," 000) = 10$] + #math.equation(block: false, alt: "81 to the power 0.25 equals the 4th root of 81 equals 3")[$81^(0.25) = root(4, 81) = 3$] ] Explain why #math.equation(block: false, alt: "x to the power the fraction 1 over 4")[$x^(frac(1, 4))$] is a reasonable notation for #math.equation(block: false, alt: "the 4th root of x")[$root(4, x)$]. \_\_\_\_\_ Explain why #math.equation(block: false, alt: "x to the power the fraction 1 over 4")[$x^(frac(1, 4))$] is a reasonable notation for #math.equation(block: false, alt: "the 4th root of x")[$root(4, x)$]. === Irrational Numbers What about #math.equation(block: false, alt: "n")[$n$]th roots such as #math.equation(block: false, alt: "the square root of 23")[$sqrt(23)$] and #math.equation(block: false, alt: "5 to the power 1 / 3")[$5^(1 / 3)$] that cannot be evaluated easily? These are examples of #strong[irrational numbers]. We can use a calculator to obtain decimal approximations for irrational numbers. For example, you can verify that #math.equation(block: true, alt: "the square root of 23 approximately equals 4.796 and 5 to the power 1 / 3 approximately equals 1.710")[$sqrt(23) ≈ 4.796 " " " and " " " 5^(1 / 3) ≈ 1.710$] It is not possible to write down an exact decimal equivalent for an irrational number, but we can find an approximation to as many decimal places as we like. #notebox("Caution", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The following keying sequence for evaluating the irrational number #math.equation(block: false, alt: "7 to the power 1 / 5")[$7^(1 / 5)$] is incorrect: 7 ^ 1 ÷ 5 ENTER You can check that this sequence calculates #math.equation(block: false, alt: "the fraction 7 to the power 1 over 5")[$display(frac(7^(1), 5))$], instead of #math.equation(block: false, alt: "7 to the power 1 / 5")[$7^(1 / 5)$]. Recall that according to the order of operations, powers are computed before multiplications or divisions. We must enclose the exponent #math.equation(block: false, alt: "1 / 5")[$1 / 5$] in parentheses and enter 7 ^ ( 1 ÷ 5 ) ENTER Or, because #math.equation(block: false, alt: "the fraction 1 over 5 equals 0.2")[$frac(1, 5) = 0.2$], we can enter 7 ^ 0.2 ENTER ] === Working with Fractional Exponents Fractional exponents simplify many calculations involving radicals. You should learn to convert easily between exponential and radical notation. Remember that a negative exponent denotes a reciprocal. #examplebox("Example 5")[][ Convert each radical to exponential notation. + #math.equation(block: false, alt: "the cube root of 12 equals 12 to the power 1 / 3")[$root(3, 12) = 12^(1 / 3)$] + #math.equation(block: false, alt: "the 4th root of 2 y equals open parenthesis 2 y close parenthesis to the power 1 / 4 or open parenthesis 2 y close parenthesis to the power 0.25")[$root(4, 2 y) = ( 2 y )^(1 / 4) " or " ( 2 y )^(0.25)$] ] Convert each radical to exponential notation. + #math.equation(block: false, alt: "the fraction 1 over the 5th root of a b equals")[$display(frac(1, root(5, a b))) =$]\_\_\_\_\_ + #math.equation(block: false, alt: "3 the 6th root of w equals")[$3 root(6, w) =$]\_\_\_\_\_ #solutionbox[ + #math.equation(block: false, alt: "open parenthesis a b close parenthesis to the power minus 1 / 5")[$( a b )^(− 1 / 5)$] + #math.equation(block: false, alt: "3 w to the power 1 / 6")[$3 w^(1 / 6)$] ] Convert each radical to exponential notation. + #math.equation(block: false, alt: "the fraction 1 over the 5th root of a b")[$display(frac(1, root(5, a b)))$] + #math.equation(block: false, alt: "3 the 6th root of w")[$3 root(6, w)$] #solutionbox[ + #math.equation(block: false, alt: "open parenthesis a b close parenthesis to the power minus 1 / 5")[$( a b )^(− 1 / 5)$] + #math.equation(block: false, alt: "3 w to the power 1 / 6")[$3 w^(1 / 6)$] ] #examplebox("Example 6")[][ Convert each power to radical notation. + #math.equation(block: false, alt: "5 to the power 1 / 2 equals the square root of 5")[$5^(1 / 2) = sqrt(5)$] + #math.equation(block: false, alt: "x to the power 0.2 equals the 5th root of x")[$x^(0.2) = root(5, x)$] + #math.equation(block: false, alt: "2 x to the power 1 / 3 equals 2 the cube root of x")[$2 x^(1 / 3) = 2 root(3, x)$] + #math.equation(block: false, alt: "8 a to the power minus 1 / 4 equals the fraction 8 over the 4th root of a")[$8 a^(− 1 / 4) = display(frac(8, root(4, a)))$] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ In Exampled, note that the exponent #math.equation(block: false, alt: "minus 1 / 4")[$− 1 / 4$] applies only to #math.equation(block: false, alt: "a")[$a$], not to #math.equation(block: false, alt: "8 a")[$8 a$]. ] + Convert #math.equation(block: false, alt: "the fraction 3 over the 4th root of 2 x")[$" " display(frac(3, root(4, 2 x))) " "$] to exponential notation. #linebreak() \_\_\_\_\_ + Convert #math.equation(block: false, alt: "minus 5 b to the power 0.125")[$" " − 5 b^(0.125) " "$] to radical notation. #linebreak() \_\_\_\_\_ #solutionbox[ + #math.equation(block: false, alt: "the fraction 3 over the 4th root of 2 x equals 3 open parenthesis 2 x close parenthesis to the power minus 1 / 4")[$display(frac(3, root(4, 2 x))) = 3 ( 2 x )^(− 1 / 4)$] + #math.equation(block: false, alt: "minus 5 b to the power 0.125 minus 5 the 8th root of b")[$− 5 b^(0.125) − 5 root(8, b)$] ] + Convert #math.equation(block: false, alt: "the fraction 3 over the 4th root of 2 x")[$display(frac(3, root(4, 2 x)))$] to exponential notation. + Convert #math.equation(block: false, alt: "minus 5 b to the power 0.125")[$− 5 b^(0.125)$] to radical notation. #solutionbox[ + #math.equation(block: false, alt: "the fraction 3 over the 4th root of 2 x equals 3 open parenthesis 2 x close parenthesis to the power minus 1 / 4")[$display(frac(3, root(4, 2 x))) = 3 ( 2 x )^(− 1 / 4)$] + #math.equation(block: false, alt: "minus 5 b to the power 0.125 minus 5 the 8th root of b")[$− 5 b^(0.125) − 5 root(8, b)$] ] Which of the following statements is true? \_\_\_\_\_ #solutionbox[ #math.equation(block: true, alt: "x to the power the fraction 1 over 3 times x to the power the fraction 1 over 3 times x to the power the fraction 1 over 3 equals x")[$x^(frac(1, 3)) ⋅ x^(frac(1, 3)) ⋅ x^(frac(1, 3)) = x$] ] Which of the following statements is true? + #math.equation(block: false, alt: "x to the power the fraction 1 over 2 plus x to the power the fraction 1 over 2 equals x")[$x^(frac(1, 2)) + x^(frac(1, 2)) = x$] + #math.equation(block: false, alt: "x to the power the fraction minus 1 over 5 equals the fraction minus 1 over x to the power 5")[$x^(frac(− 1, 5)) = display(frac(− 1, x^(5)))$] + #math.equation(block: false, alt: "x to the power the fraction 1 over 4 times x to the power 4 equals x")[$x^(frac(1, 4)) ⋅ x^(4) = x$] + #math.equation(block: false, alt: "x to the power the fraction 1 over 3 times x to the power the fraction 1 over 3 times x to the power the fraction 1 over 3 equals x")[$x^(frac(1, 3)) ⋅ x^(frac(1, 3)) ⋅ x^(frac(1, 3)) = x$] === Using Fractional Exponents to Solve Equations In Chapter 2, we learned that raising to powers and taking roots are inverse operations, that is, each operation undoes the effects of the other. This relationship is especially easy to see when the root is denoted by a fractional exponent. For example, to solve the equation The sentence above, kept live for x ≥ 0: the blue curve is the power function y = xⁿ and the red curve is the root function y = x^(1/n), with the index n on a slider. The two curves are mirror images across the dashed line y = x, because taking the nth root undoes raising to the nth power — reflecting a graph across that line is what “undoing” looks like. Whatever n you choose, both curves pass through (0, 0) and meet again at (1, 1), since 0 and 1 are their own nth powers and their own nth roots. Raise n and watch the pair pull apart: the power curve hugs the x-axis longer and then climbs more steeply, and its mirror the root curve does the reverse, climbing steeply at first and then flattening out.{"functions":\[{"color":"\#6a6a6a","domain":\[0,8\],"expression":"x","expression2":"","id":"mirror","inequality":"lt","lineDash":\[6,4\],"lineWidth":1.5,"mode":"cartesian","tMax":10,"tMin":-10,"variable":"x","visible":true},{"color":"\#1f4e79","domain":\[0,6\],"expression":"x^n","expression2":"","id":"power","inequality":"lt","lineDash":\[\],"lineWidth":2.5,"mode":"cartesian","tMax":10,"tMin":-10,"variable":"x","visible":true},{"color":"\#b8442a","expression":"t^n","expression2":"t","id":"root","inequality":"lt","lineDash":\[\],"lineWidth":2.5,"mode":"parametric","tMax":2.4,"tMin":0,"variable":"t","visible":true}\],"grid":{"majorSpacing":1,"showAxes":true,"showLabels":true},"parameters":\[{"id":"p-n","label":"Index n (square, cube, fourth, fifth)","max":5,"min":2,"name":"n","step":1,"value":2}\],"title":"y = x\\u207f and y = x^(1/n), mirrored in y = x","version":1,"viewport":{"centerX":2,"centerY":2,"scale":60}}#math.equation(block: true, alt: "x to the power 4 equals 250")[$x^(4) = 250$] we would take the fourth root of each side. But instead of using radical notation, we can raise both sides of the equation to the power #math.equation(block: false, alt: "the fraction 1 over 4")[$display(frac(1, 4))$]: #math.equation(block: true, alt: "open parenthesis x to the power 4 close parenthesis to the power 1 / 4 equals 250 to the power 1 / 4")[$attach(( x^(4) ), t: 1 / 4) = 250^(1 / 4)$] The third law of exponents tells us that #math.equation(block: false, alt: "open parenthesis x to the power a close parenthesis to the power b equals x to the power a b")[$attach(( x^(a) ), t: b) = x^(a b)$], so #math.equation(block: true, alt: "open parenthesis x to the power 4 close parenthesis to the power 1 / 4 equals x to the power open parenthesis 1 / 4 close parenthesis open parenthesis 4 close parenthesis equals x to the power 1")[$attach(( x^(4) ), t: 1 / 4) = x^(( 1 / 4 ) ( 4 )) = x^(1)$] We evaluate the right side to find #math.equation(block: false, alt: "250 to the power 1 / 4")[$250^(1 / 4)$] or #math.equation(block: false, alt: "the 4th root of 250 approximately equals 3.98")[$root(4, 250) ≈ 3.98$]. In general, to solve an equation involving a power function #math.equation(block: false, alt: "x to the power n")[$x^(n)$], we first isolate the power, then raise both sides to the exponent #math.equation(block: false, alt: "the fraction 1 over n")[$display(frac(1, n))$]. #examplebox("Example 7")[][ For astronomers, the mass of a star is its most important property, but it is also the most difficult to measure directly. For many stars, their luminosity, or brightness, varies roughly as the fourth power of the mass. + Our Sun has luminosity #math.equation(block: false, alt: "4 times 10 to the power 26")[$4 × 10^(26)$] watts and mass #math.equation(block: false, alt: "2 times 10 to the power 30")[$2 × 10^(30)$] kilograms. Because the numbers involved are so large, astronomers often use these solar constants as units of measure: #linebreak() The luminosity of the Sun is #math.equation(block: false, alt: "1")[$1$] solar luminosity, and its mass is #math.equation(block: false, alt: "1")[$1$] solar mass. #linebreak() Write a power function for the luminosity, #math.equation(block: false, alt: "L")[$L$], of a star in terms of its mass, #math.equation(block: false, alt: "M")[$M$], using units of solar mass and solar luminosity. + The star Sirius is #math.equation(block: false, alt: "23")[$23$] times brighter than the Sun, so its luminosity is #math.equation(block: false, alt: "23")[$23$] solar luminosities. Estimate the mass of Sirius in units of solar mass. #solutionbox[ + Because #math.equation(block: false, alt: "L")[$L$] varies as the fourth power of #math.equation(block: false, alt: "M")[$M$], we have #math.equation(block: true, alt: "L equals k M to the power 4")[$L = k M^(4)$] Substituting the values of #math.equation(block: false, alt: "L")[$L$] and #math.equation(block: false, alt: "M")[$M$] for the Sun (namely, #math.equation(block: false, alt: "L equals 1")[$L = 1$] and #math.equation(block: false, alt: "M equals 1")[$M = 1$]), we find #math.equation(block: true, alt: "1 equals k open parenthesis 1 close parenthesis to the power 4")[$1 = k ( 1 )^(4)$] so #math.equation(block: false, alt: "k equals 1")[$k = 1$] and #math.equation(block: false, alt: "L equals M to the power 4")[$L = M^(4)$]. + We substitute the luminosity of Sirius, #math.equation(block: false, alt: "L equals 23")[$L = 23$], to get #math.equation(block: true, alt: "23 equals M to the power 4")[$23 = M^(4)$] To solve the equation for #math.equation(block: false, alt: "M")[$M$], we raise both sides to the #math.equation(block: false, alt: "the fraction 1 over 4")[$display(frac(1, 4))$] power. #math.equation(block: true, alt: "open parenthesis 23 close parenthesis to the power 1 / 4, equals open parenthesis M to the power 4 close parenthesis to the power 1 / 4; 2.1899, equals M")[$( 23 )^(1 / 4) & = attach(( M^(4) ), t: 1 / 4) \ 2.1899 & = M$] The mass of Sirius is about #math.equation(block: false, alt: "2.2")[$2.2$] solar masses, or about #math.equation(block: false, alt: "2.2")[$2.2$] times the mass of the Sun. ] ] A spherical fish tank in the lobby of the Atlantis Hotel holds about 905 cubic feet of water. What is the radius of the fish tank? Answer: About \_\_\_\_\_ feet #solutionbox[ Solve #math.equation(block: false, alt: "the fraction 4 over 3 π r cubed equals 905")[$display(frac(4, 3)) π r^(3) = 905$] to find that the radius #math.equation(block: false, alt: "r")[$r$] is about #math.equation(block: false, alt: "6")[$6$] feet ] A spherical fish tank in the lobby of the Atlantis Hotel holds about 905 cubic feet of water. What is the radius of the fish tank? #solutionbox[ Solve #math.equation(block: false, alt: "the fraction 4 over 3 π r cubed equals 905")[$" " display(frac(4, 3)) π r^(3) = 905 " "$] to find that the radius #math.equation(block: false, alt: "r")[$r$] is about #math.equation(block: false, alt: "6")[$6$] feet. ] === Power Functions The basic functions #math.equation(block: false, alt: "y equals the square root of x")[$y = sqrt(x)$] and #math.equation(block: false, alt: "y equals the cube root of x")[$y = root(3, x)$] are power functions of the form #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x to the power 1 / n")[$f ( x ) = x^(1 / n)$], and the graphs of all such functions have shapes similar to those two, depending on whether the index of the root is even or odd. Figure (a) shows the graphs of #math.equation(block: true, alt: "y equals x to the power 1 / 2 , y equals x to the power 1 / 4 , and y equals x to the power 1 / 6")[$y = x^(1 / 2) , " " " " y = x^(1 / 4) , " " " " "and" " " " " y = x^(1 / 6)$] Figure (b) shows the graphs of #math.equation(block: true, alt: "y equals x to the power 1 / 3 , y equals x to the power 1 / 5 , and y equals x to the power 1 / 7")[$y = x^(1 / 3) , " " " " y = x^(1 / 5) , " " " " "and" " " " " y = x^(1 / 7)$] #figure(figph[graphs of roots], alt: "graphs of roots", caption: none) We cannot take an even root of a negative number. (See A Note on Roots of Negative Numbers at the end of this section.) Hence, if #math.equation(block: false, alt: "n")[$n$] is even, the domain of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x to the power 1 / n")[$f ( x ) = x^(1 / n)$] is restricted to nonnegative real numbers, but if #math.equation(block: false, alt: "n")[$n$] is odd, the domain of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x to the power 1 / n")[$f ( x ) = x^(1 / n)$] is the set of all real numbers. We will also encounter power functions with negative exponents. For example, an animal's heart rate is related to its size or mass, with smaller animals generally having faster heart rates. The heart rates of mammals are given approximately by the power function #math.equation(block: true, alt: "H open parenthesis m close parenthesis equals k m to the power minus 1 / 4")[$H ( m ) = k m^(− 1 / 4)$] where #math.equation(block: false, alt: "m")[$m$] is the animal's mass and #math.equation(block: false, alt: "k")[$k$] is a constant. #examplebox("Example 8")[][ A typical human male weighs about #math.equation(block: false, alt: "70")[$70$] kilograms and has a resting heart rate of #math.equation(block: false, alt: "70")[$70$] beats per minute. + Find the constant of proportionality, #math.equation(block: false, alt: "k")[$k$], and write a formula for #math.equation(block: false, alt: "H open parenthesis m close parenthesis")[$H ( m )$]. + Fill in the table with the heart rates of the mammals whose masses are given.#figure(table( columns: 9, align: left, inset: 6pt, table.header([Animal], [Shrew], [Rabbit], [Cat], [Wolf], [Horse], [Polar bear], [Elephant], [Whale]), [Mass (kg)], [#math.equation(block: false, alt: "0.004")[$0.004$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "80")[$80$]], [#math.equation(block: false, alt: "300")[$300$]], [#math.equation(block: false, alt: "600")[$600$]], [#math.equation(block: false, alt: "5400")[$5400$]], [#math.equation(block: false, alt: "70 , 000")[$70 , 000$]], [Heart rate], [], [], [], [], [], [], [], [], )) + Sketch a graph of #math.equation(block: false, alt: "H")[$H$] for masses up to #math.equation(block: false, alt: "6000")[$6000$] kilograms. #solutionbox[ + We substitute #math.equation(block: false, alt: "H equals 70")[$H = 70$] and #math.equation(block: false, alt: "m equals 70")[$m = 70$] into the equation; then solve for #math.equation(block: false, alt: "k")[$k$]. #math.equation(block: true, alt: "70, equals k times 70 to the power minus 1 / 4; k, equals the fraction 70 over 70 to the power minus 1 / 4 equals 70 to the power 5 / 4 approximately equals 202.5")[$70 & = k ⋅ 70^(− 1 / 4) \ k & = frac(70, 70^(− 1 / 4)) = 70^(5 / 4) ≈ 202.5$] Thus, #math.equation(block: false, alt: "H open parenthesis m close parenthesis equals 202.5 m to the power minus 1 / 4")[$H ( m ) = 202.5 m^(− 1 / 4)$]. + We evaluate the function #math.equation(block: false, alt: "H")[$H$] for each of the masses given in the table.#figure(table( columns: 9, align: left, inset: 6pt, table.header([Animal], [Shrew], [Rabbit], [Cat], [Wolf], [Horse], [Polar bear], [Elephant], [Whale]), [Mass (kg)], [#math.equation(block: false, alt: "0.004")[$0.004$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "80")[$80$]], [#math.equation(block: false, alt: "300")[$300$]], [#math.equation(block: false, alt: "600")[$600$]], [#math.equation(block: false, alt: "5400")[$5400$]], [#math.equation(block: false, alt: "70 , 000")[$70 , 000$]], [Heart rate], [#math.equation(block: false, alt: "805")[$805$]], [#math.equation(block: false, alt: "170")[$170$]], [#math.equation(block: false, alt: "143")[$143$]], [#math.equation(block: false, alt: "68")[$68$]], [#math.equation(block: false, alt: "49")[$49$]], [#math.equation(block: false, alt: "41")[$41$]], [#math.equation(block: false, alt: "24")[$24$]], [#math.equation(block: false, alt: "12")[$12$]], )) + We plot the points in the table to obtain the graph shown below.#figure(figph[graph of heart rate vs mass], alt: "graph of heart rate vs mass", caption: none) ] ] Many properties relating to the growth of plants and animals can be described by power functions of their mass. The study of the relationship between the growth rates of different parts of an organism, or of organisms of similar type, is called #strong[allometry]. An equation of the form #math.equation(block: true, alt: "variable equals k open parenthesis mass close parenthesis to the power p")[$"variable" = k ( "mass" )^(p)$] used to describe such a relationship is called an #strong[allometric equation]. Of course, power functions can be expressed using any of the notations we have discussed. For example, the function in Example can be written as #math.equation(block: true, alt: "H open parenthesis m close parenthesis equals 202.5 m to the power minus 1 / 4 or H open parenthesis m close parenthesis equals 202.5 m to the power minus 0.25 or H open parenthesis m close parenthesis equals the fraction 202.5 over the 4th root of m")[$H ( m ) = 202.5 m^(− 1 / 4) " " " " " " " or " " " " " " " H ( m ) = 202.5 m^(− 0.25) " " " " " " " or" " " " " " " H ( m ) = frac(202.5, root(4, m))$] + Complete the table of values for the power function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x to the power minus 1 / 2")[$f ( x ) = x^(− 1 / 2)$].#figure(table( columns: 6, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "0.1")[$0.1$]], [#math.equation(block: false, alt: "0.25")[$0.25$]], [#math.equation(block: false, alt: "0.5")[$0.5$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]]), [#math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]], [\_\_\_\_\_], [\_\_\_\_\_], [\_\_\_\_\_], [\_\_\_\_\_], [\_\_\_\_\_], )) #figure(table( columns: 6, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "200")[$200$]]), [#math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]], [\_\_\_\_\_], [\_\_\_\_\_], [\_\_\_\_\_], [\_\_\_\_\_], [\_\_\_\_\_], )) + Sketch the graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$]. + Write the formula for #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] with a decimal exponent, and with radical notation. #linebreak() #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] is #linebreak() \_\_\_\_\_ #linebreak() and also, #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] is #linebreak() \_\_\_\_\_ #solutionbox[ + #figure(table( columns: 6, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "0.1")[$0.1$]], [#math.equation(block: false, alt: "0.25")[$0.25$]], [#math.equation(block: false, alt: "0.5")[$0.5$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]]), [#math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]], [#math.equation(block: false, alt: "3.2")[$3.2$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "1.4")[$1.4$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "0.71")[$0.71$]], )) #figure(table( columns: 6, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "200")[$200$]]), [#math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]], [#math.equation(block: false, alt: "0.5")[$0.5$]], [#math.equation(block: false, alt: "0.35")[$0.35$]], [#math.equation(block: false, alt: "0.32")[$0.32$]], [#math.equation(block: false, alt: "0.22")[$0.22$]], [#math.equation(block: false, alt: "0.1")[$0.1$]], )) + A graph is below. + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x to the power minus 0.5")[$f ( x ) = x^(− 0.5)$], #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the fraction 1 over the square root of x")[$f ( x ) = display(frac(1, sqrt(x)))$] ] #figure(figph[power function], alt: "power function", caption: none) + Complete the table of values for the power function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x to the power minus 1 / 2")[$" " f ( x ) = x^(− 1 / 2)$].#figure(table( columns: 6, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "0.1")[$0.1$]], [#math.equation(block: false, alt: "0.25")[$0.25$]], [#math.equation(block: false, alt: "0.5")[$0.5$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]]), [#math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], )) #figure(table( columns: 6, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "200")[$200$]]), [#math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], )) + Sketch the graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$]. + Write the formula for #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] with a decimal exponent, and with radical notation. #solutionbox[ + #figure(table( columns: 6, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "0.1")[$0.1$]], [#math.equation(block: false, alt: "0.25")[$0.25$]], [#math.equation(block: false, alt: "0.5")[$0.5$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]]), [#math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]], [#math.equation(block: false, alt: "3.2")[$3.2$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "1.4")[$1.4$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "0.71")[$0.71$]], )) #figure(table( columns: 6, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "200")[$200$]]), [#math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]], [#math.equation(block: false, alt: "0.5")[$0.5$]], [#math.equation(block: false, alt: "0.35")[$0.35$]], [#math.equation(block: false, alt: "0.32")[$0.32$]], [#math.equation(block: false, alt: "0.22")[$0.22$]], [#math.equation(block: false, alt: "0.1")[$0.1$]], )) + #figure(figph[power function], alt: "power function", caption: none) + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x to the power minus 0.5")[$f ( x ) = x^(− 0.5)$], #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the fraction 1 over the square root of x")[$f ( x ) = display(frac(1, sqrt(x)))$] ] Describe the differences in the graphs of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x to the power the fraction 1 over n")[$f ( x ) = x^(frac(1, n))$] for #math.equation(block: false, alt: "n")[$n$] positive and negative, for #math.equation(block: false, alt: "x greater than 0")[$x > 0$]. \_\_\_\_\_ Describe the differences in the graphs of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x to the power the fraction 1 over n")[$f ( x ) = x^(frac(1, n))$] for #math.equation(block: false, alt: "n")[$n$] positive and negative, for #math.equation(block: false, alt: "x greater than 0")[$x > 0$]. === Solving Radical Equations A #strong[radical equation] is one in which the variable appears under a square root or other radical. The radical may be denoted by a fractional exponent. For example, the equation #math.equation(block: true, alt: "5 x to the power 1 / 3 equals 32")[$5 x^(1 / 3) = 32$] is a radical equation because #math.equation(block: false, alt: "x to the power 1 / 3 equals the cube root of x")[$x^(1 / 3) = root(3, x)$]. To solve the equation, we first isolate the power to get #math.equation(block: true, alt: "x to the power 1 / 3 equals 6.4")[$x^(1 / 3) = 6.4$] Then we raise both sides of the equation to the reciprocal of #math.equation(block: false, alt: "the fraction 1 over 3")[$display(frac(1, 3))$], or #math.equation(block: false, alt: "3")[$3$]. #math.equation(block: true, alt: "open parenthesis x to the power 1 / 3 close parenthesis cubed, equals 6.4 cubed; x, equals 262.144")[$attach(( x^(1 / 3) ), t: 3) & = 6.4^(3) \ x & = 262.144$] #examplebox("Example 9")[][ When a car brakes suddenly, its speed can be estimated from the length of the skid marks it leaves on the pavement. A formula for the car’s speed, in miles per hour, is #math.equation(block: false, alt: "v equals f open parenthesis d close parenthesis equals open parenthesis 24 d close parenthesis to the power 1 / 2")[$v = f ( d ) = ( 24 d )^(1 / 2)$], where the length of the skid marks, #math.equation(block: false, alt: "d")[$d$], is given in feet. + If a car leaves skid marks #math.equation(block: false, alt: "80")[$80$] feet long, how fast was the car traveling when the driver applied the brakes? + How far will a car skid if its driver applies the brakes while traveling #math.equation(block: false, alt: "80")[$80$] miles per hour? #solutionbox[ + To find the velocity of the car, we evaluate the function for #math.equation(block: false, alt: "d equals 80")[$d = 80$]. #math.equation(block: true, alt: "v, equals open parenthesis 24 times 80 close parenthesis to the power 1 / 2; equals open parenthesis 1920 close parenthesis to the power 1 / 2 approximately equals 43.8178046")[$v & = ( 24 ⋅ 80 )^(1 / 2) \ & = ( 1920 )^(1 / 2) ≈ 43.8178046$] The car was traveling at approximately #math.equation(block: false, alt: "44")[$44$] miles per hour. + We would like to find the value of #math.equation(block: false, alt: "d")[$d$] when the value of #math.equation(block: false, alt: "v")[$v$] is known. We substitute #math.equation(block: false, alt: "v equals 80")[$v = 80$] into the formula and solve the equation #math.equation(block: true, alt: "80 equals open parenthesis 24 d close parenthesis to the power 1 / 2 Solve for d .")[$80 = ( 24 d )^(1 / 2) " " " " " Solve for " bold(italic(d)) .$] Because #math.equation(block: false, alt: "d")[$d$] appears to the power #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$], we first square both sides of the equation to get #math.equation(block: true, alt: "80 squared, equals open parenthesis open parenthesis 24 d close parenthesis to the power 1 / 2 close parenthesis squared, Square both sides.; 6400, equals 24 d, Divide by 24.; 266.6 bar, equals d")[$80^(2) & = attach(( ( 24 d )^(1 / 2) ), t: 2) & & "Square both sides." \ 6400 & = 24 d & & "Divide by " 24. \ 266. limits(6)^(―) & = d$] You can check that this value for #math.equation(block: false, alt: "d")[$d$] works in the original equation. Thus, the car will skid approximately #math.equation(block: false, alt: "267")[$267$] feet. A graph of the function #math.equation(block: false, alt: "v equals open parenthesis 24 d close parenthesis to the power 1 / 2")[$v = ( 24 d )^(1 / 2)$] is shown below, along with the points corresponding to the values in parts (a) and (b).#figure(figph[velocity vs braking distance], alt: "velocity vs braking distance", caption: none) ] ] Which law of exponents do we use when solving #math.equation(block: false, alt: "24 d to the power the fraction 1 over 2 equals 80")[$24 d^(frac(1, 2)) = 80$]? \_\_\_\_\_ #solutionbox[ #math.equation(block: true, alt: "open parenthesis x to the power n close parenthesis to the power m equals x to the power n m")[$( x^(n) )^(m) = x^(n m)$] ] Which law of exponents do we use when solving #math.equation(block: false, alt: "24 d to the power the fraction 1 over 2 equals 80")[$24 d^(frac(1, 2)) = 80$]? + #math.equation(block: false, alt: "open parenthesis x plus y close parenthesis to the power n equals x to the power n plus y to the power n")[$( x + y )^(n) = x^(n) + y^(n)$] + #math.equation(block: false, alt: "open parenthesis x to the power n close parenthesis to the power m equals x to the power n m")[$( x^(n) )^(m) = x^(n m)$] + #math.equation(block: false, alt: "open parenthesis x y close parenthesis to the power n equals x to the power n y to the power n")[$( x y )^(n) = x^(n) y^(n)$] + #math.equation(block: false, alt: "x to the power n x to the power m equals x to the power n plus m")[$x^(n) x^(m) = x^(n + m)$] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Thus, we can solve an equation where one side is an #math.equation(block: false, alt: "n")[$n$]th root of #math.equation(block: false, alt: "x")[$x$] by raising both sides of the equation to the #math.equation(block: false, alt: "n")[$n$]th power. We must be careful when raising both sides of an equation to an even power, since extraneous solutions may be introduced. However, because most applications of power functions deal with positive domains only, they do not usually involve extraneous solutions. ] In Example, we found the heart-rate function, #math.equation(block: false, alt: "H open parenthesis m close parenthesis equals 202.5 m to the power minus 1 / 4")[$H ( m ) = 202.5 m^(− 1 / 4)$]. What would be the mass of an animal whose heart rate is #math.equation(block: false, alt: "120")[$120$] beats per minute? Answer: \_\_\_\_\_ kg #solutionbox[ We solve #math.equation(block: false, alt: "120 equals 202.5 m to the power minus 1 / 4")[$120 = 202.5 m^(− 1 / 4)$] to find #math.equation(block: false, alt: "m approximately equals 8.1")[$m ≈ 8.1$] kg ] What would be the mass of an animal whose heart rate is #math.equation(block: false, alt: "120")[$120$] beats per minute? #solutionbox[ We solve #math.equation(block: false, alt: "120 equals 202.5 m to the power minus 1 / 4")[$" " 120 = 202.5 m^(− 1 / 4) " "$] to find #math.equation(block: false, alt: "m approximately equals 8.1")[$m ≈ 8.1$] kg. ] === A Note on Roots of Negative Numbers You already know that #math.equation(block: false, alt: "the square root of minus 9")[$sqrt(− 9)$] is not a real number, because there is no real number whose square is #math.equation(block: false, alt: "minus 9")[$− 9$]. Similarly, #math.equation(block: false, alt: "the 4th root of minus 16")[$root(4, − 16)$] is not a real number, because there is no real number #math.equation(block: false, alt: "r")[$r$] for which #math.equation(block: false, alt: "r to the power 4 equals minus 16")[$r^(4) = − 16$]. (Both of these radicals are #strong[complex numbers]. Complex numbers are discussed in Chapter 7.) In general, we cannot find an even root (square root, fourth root, and so on) of a negative number. On the other hand, every positive number has two even roots that are real numbers. For example, both #math.equation(block: false, alt: "3")[$3$] and #math.equation(block: false, alt: "minus 3")[$− 3$] are square roots of #math.equation(block: false, alt: "9")[$9$]. The symbol #math.equation(block: false, alt: "the square root of 9")[$sqrt(9)$] refers only to the positive, or #strong[principal root], of #math.equation(block: false, alt: "9")[$9$]. If we want to refer to the negative square root of #math.equation(block: false, alt: "9")[$9$], we must write #math.equation(block: false, alt: "minus the square root of 9 equals minus 3")[$− sqrt(9) = − 3$]. Similarly, both #math.equation(block: false, alt: "2")[$2$] and #math.equation(block: false, alt: "minus 2")[$− 2$] are fourth roots of #math.equation(block: false, alt: "16")[$16$], because #math.equation(block: false, alt: "2 to the power 4 equals 16")[$2^(4) = 16$] and #math.equation(block: false, alt: "open parenthesis minus 2 close parenthesis to the power 4 equals 16")[$( − 2 )^(4) = 16$]. However, the symbol #math.equation(block: false, alt: "the 4th root of 16")[$root(4, 16)$] refers to the principal, or positive, fourth root only. Thus, #math.equation(block: true, alt: "the 4th root of 16 equals 2 and minus the 4th root of 16 equals minus 2")[$root(4, 16) = 2 " " " " " and " " " " " − root(4, 16) = − 2$] Things are simpler for odd roots (cube roots, fifth roots, and so on). Every real number, whether positive, negative, or zero, has exactly one real-valued odd root. For example, #math.equation(block: true, alt: "the 5th root of 32 equals 2 and the 5th root of minus 32 equals minus 2")[$root(5, 32) = 2 " " " " " and " " " " " root(5, − 32) = − 2$] Here is a summary of our discussion. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Roots of Real Numbers] + Every positive number has two real-valued roots, one positive and one negative, if the index is even. + A negative number has no real-valued root if the index is even. + Every real number, positive, negative, or zero, has exactly one real-valued root if the index is odd. ] #examplebox("Example 10")[][ + #math.equation(block: false, alt: "the 4th root of minus 625")[$root(4, − 625)$] is not a real number. + #math.equation(block: false, alt: "minus the 4th root of 625 equals minus 5")[$− root(4, 625) = − 5$] + #math.equation(block: false, alt: "the 5th root of minus 1 equals minus 1")[$root(5, − 1) = − 1$] + #math.equation(block: false, alt: "the 4th root of minus 1")[$root(4, − 1)$] is not a real number. ] The same principles apply to powers with fractional exponents. Thus #math.equation(block: true, alt: "open parenthesis minus 32 close parenthesis to the power 1 / 5 equals minus 2")[$( − 32 )^(1 / 5) = − 2$] but #math.equation(block: false, alt: "open parenthesis minus 64 close parenthesis to the power 1 / 6")[$( − 64 )^(1 / 6)$] is not a real number. On the other hand, #math.equation(block: true, alt: "minus 64 to the power 1 / 6 equals minus 2")[$− 64^(1 / 6) = − 2$] because the exponent #math.equation(block: false, alt: "1 / 6")[$1 / 6$] applies only to #math.equation(block: false, alt: "64")[$64$], and the negative sign is applied after the root is computed. Which of the following is undefined for negative #math.equation(block: false, alt: "x")[$x$]? \_\_\_\_\_ #solutionbox[ #math.equation(block: false, alt: "x to the power the fraction 1 over 4")[$x^(frac(1, 4))$] is undefined for negative #math.equation(block: false, alt: "x")[$x$]. ] Which of the following is undefined for negative #math.equation(block: false, alt: "x")[$x$]? + #math.equation(block: false, alt: "x to the power the fraction 1 over 4")[$x^(frac(1, 4))$] + #math.equation(block: false, alt: "x to the power minus 4")[$x^(− 4)$] + #math.equation(block: false, alt: "open parenthesis minus x close parenthesis to the power the fraction 1 over 4")[$( − x )^(frac(1, 4))$] + #math.equation(block: false, alt: "open parenthesis minus x close parenthesis to the power minus 4")[$( − x )^(− 4)$] Evaluate each power, if possible. Enter "DNE" if it is not possible to evaluate. + #math.equation(block: false, alt: "minus 81 to the power 1 / 4 equals")[$− 81^(1 / 4) =$]\_\_\_\_\_ + #math.equation(block: false, alt: "open parenthesis minus 81 close parenthesis to the power 1 / 4 equals")[$( − 81 )^(1 / 4) =$]\_\_\_\_\_ + #math.equation(block: false, alt: "minus 64 to the power 1 / 3 equals")[$− 64^(1 / 3) =$]\_\_\_\_\_ + #math.equation(block: false, alt: "open parenthesis minus 64 close parenthesis to the power 1 / 3 equals")[$( − 64 )^(1 / 3) =$]\_\_\_\_\_ #solutionbox[ + #math.equation(block: false, alt: "minus 3")[$− 3$] + undefined + #math.equation(block: false, alt: "minus 4")[$− 4$] + #math.equation(block: false, alt: "minus 4")[$− 4$] ] Evaluate each power, if possible. Enter "DNE" if it is not possible to evaluate. + #math.equation(block: false, alt: "minus 81 to the power 1 / 4")[$− 81^(1 / 4)$] + #math.equation(block: false, alt: "open parenthesis minus 81 close parenthesis to the power 1 / 4")[$( − 81 )^(1 / 4)$] + #math.equation(block: false, alt: "minus 64 to the power 1 / 3")[$− 64^(1 / 3)$] + #math.equation(block: false, alt: "open parenthesis minus 64 close parenthesis to the power 1 / 3")[$( − 64 )^(1 / 3)$] #solutionbox[ + #math.equation(block: false, alt: "minus 3")[$− 3$] + undefined + #math.equation(block: false, alt: "minus 4")[$− 4$] + #math.equation(block: false, alt: "minus 4")[$− 4$] ] What is the domain of the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x to the power the fraction 1 over 2 n")[$f ( x ) = x^(frac(1, 2 n))$], and why? \_\_\_\_\_ What is the domain of the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x to the power the fraction 1 over 2 n")[$f ( x ) = x^(frac(1, 2 n))$], and why? === Section Summary ==== Vocabulary Look up the definitions of new terms in the Glossary. - #math.equation(block: false, alt: "n")[$n$]th root - Radical - Radical equation - Radical notation - Index - Exponential notation - Radicand - Allometric equation - Irrational number ==== CONCEPTS + #math.equation(block: false, alt: "n")[$n$]th roots: #math.equation(block: false, alt: "s")[$s$] is called an #strong[#math.equation(block: false, alt: "n")[$n$]th root of #math.equation(block: false, alt: "b")[$b$]] if #math.equation(block: false, alt: "s to the power n equals b")[$s^(n) = b$]. + Exponential notation: For any integer #math.equation(block: false, alt: "n greater than or equal to 2")[$n ≥ 2$] and for #math.equation(block: false, alt: "a greater than or equal to 0")[$a ≥ 0$], #math.equation(block: false, alt: "a to the power 1 / n equals the nth root of a")[$a^(1 / n) = root(n, a)$]. + We cannot write down an exact decimal equivalent for an #strong[irrational number], but we can approximate an irrational number to as many decimal places as we like. + We can solve the equation #math.equation(block: false, alt: "x to the power n equals b")[$x^(n) = b$] by raising both sides to the #math.equation(block: false, alt: "the fraction 1 over n")[$display(frac(1, n))$] power. + An #strong[allometric equation] is a power function of the form #math.equation(block: false, alt: "variable equals k open parenthesis mass close parenthesis to the power p")[$"variable" = k ( "mass" )^(p)$]. + We can solve the equation #math.equation(block: false, alt: "x to the power 1 / n equals b")[$x^(1 / n) = b$] by raising both sides to the #math.equation(block: false, alt: "n")[$n$]th power. + #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Roots of Real Numbers] - Every positive number has two real-valued roots, one positive and one negative, if the index is even. - A negative number has no real-valued root if the index is even. - Every real number, positive, negative, or zero, has exactly one real-valued root if the index is odd. ] ==== STUDY QUESTIONS + Use an example to illustrate the terms radical, radicand, index, and principal root. + Explain why #math.equation(block: false, alt: "x to the power 1 / 4")[$x^(1 / 4)$] is a reasonable notation for #math.equation(block: false, alt: "the 4th root of x")[$root(4, x)$]. + What does the notation #math.equation(block: false, alt: "x to the power 0.2")[$x^(0.2)$] mean? + Express each of the following algebraic notations in words; then evaluate each for #math.equation(block: false, alt: "equals 16")[$= 16$]: #math.equation(block: true, alt: "4 x , x to the power 4 , the fraction x over 4 , the fraction 1 over 4 x , x to the power 1 / 4 , x minus 4 , x to the power minus 1 / 4")[$4 x , " " " " x^(4) , " " " " frac(x, 4) , " " " " frac(1, 4) x , " " " " x^(1 / 4) , " " " " x − 4 , " " " " x^(− 1 / 4)$] + How is the third law of exponents, #math.equation(block: false, alt: "open parenthesis x to the power a close parenthesis to the power b equals x to the power a b ,")[$( x^(a) )^(b) = x^(a b) ,$] useful in solving equations? ==== SKILLS Practice each skill in the Homework problems listed. + Evaluate powers and roots: \#1–8, 17–20 + Convert between radical and exponential notation: \#9–16, 21 and 22 + Solve radical equations: \#23–38, 59 and 60 + Graph and analyze power functions: \#39–58 + Work with fractional exponents: \#61–68 === Homework 3.3 For Problems 1–4, find the indicated root without using a calculator; then check your answers. + #math.equation(block: false, alt: "the square root of 121")[$sqrt(121)$] + #math.equation(block: false, alt: "the cube root of 27")[$root(3, 27)$] + #math.equation(block: false, alt: "the 4th root of 625")[$root(4, 625)$] + #math.equation(block: false, alt: "11")[$11$] + #math.equation(block: false, alt: "3")[$3$] + #math.equation(block: false, alt: "5")[$5$] + #math.equation(block: false, alt: "the square root of 169")[$sqrt(169)$] + #math.equation(block: false, alt: "the cube root of 64")[$root(3, 64)$] + #math.equation(block: false, alt: "the 4th root of 81")[$root(4, 81)$] + #math.equation(block: false, alt: "the 5th root of 32")[$root(5, 32)$] + #math.equation(block: false, alt: "the 4th root of 16")[$root(4, 16)$] + #math.equation(block: false, alt: "the cube root of 729")[$root(3, 729)$] + #math.equation(block: false, alt: "2")[$2$] + #math.equation(block: false, alt: "2")[$2$] + #math.equation(block: false, alt: "9")[$9$] + #math.equation(block: false, alt: "the 5th root of 100 , 000")[$root(5, 100 "," 000)$] + #math.equation(block: false, alt: "the 4th root of 1296")[$root(4, 1296)$] + #math.equation(block: false, alt: "the cube root of 343")[$root(3, 343)$] For Problems 5–8, find the indicated power without using a calculator; then check your answers. + #math.equation(block: false, alt: "9 to the power 1 / 2")[$9^(1 / 2)$] + #math.equation(block: false, alt: "81 to the power 1 / 4")[$81^(1 / 4)$] + #math.equation(block: false, alt: "64 to the power 1 / 6")[$64^(1 / 6)$] + #math.equation(block: false, alt: "3")[$3$] + #math.equation(block: false, alt: "3")[$3$] + #math.equation(block: false, alt: "2")[$2$] + #math.equation(block: false, alt: "25 to the power 1 / 2")[$25^(1 / 2)$] + #math.equation(block: false, alt: "16 to the power 1 / 4")[$16^(1 / 4)$] + #math.equation(block: false, alt: "27 to the power 1 / 3")[$27^(1 / 3)$] + #math.equation(block: false, alt: "32 to the power 0.2")[$32^(0.2)$] + #math.equation(block: false, alt: "8 to the power minus 1 / 3")[$8^(− 1 / 3)$] + #math.equation(block: false, alt: "64 to the power minus 0.5")[$64^(− 0.5)$] + #math.equation(block: false, alt: "2")[$2$] + #math.equation(block: false, alt: "the fraction 1 over 2")[$display(frac(1, 2))$] + #math.equation(block: false, alt: "the fraction 1 over 8")[$display(frac(1, 8))$] + #math.equation(block: false, alt: "625 to the power 0.25")[$625^(0.25)$] + #math.equation(block: false, alt: "243 to the power minus 1 / 5")[$243^(− 1 / 5)$] + #math.equation(block: false, alt: "49 to the power minus 0.5")[$49^(− 0.5)$] For Problems 9–12, write each expression in radical form. + #math.equation(block: false, alt: "3 to the power 1 / 2")[$3^(1 / 2)$] + #math.equation(block: false, alt: "4 x to the power 1 / 3")[$4 x^(1 / 3)$] + #math.equation(block: false, alt: "open parenthesis 4 x close parenthesis to the power 0.2")[$( 4 x )^(0.2)$] + #math.equation(block: false, alt: "the square root of 3")[$sqrt(3)$] + #math.equation(block: false, alt: "4 the cube root of x")[$4 root(3, x)$] + #math.equation(block: false, alt: "the 5th root of 4 x")[$root(5, 4 x)$] + #math.equation(block: false, alt: "7 to the power 1 / 2")[$7^(1 / 2)$] + #math.equation(block: false, alt: "3 x to the power 1 / 4")[$3 x^(1 / 4)$] + #math.equation(block: false, alt: "open parenthesis 3 x close parenthesis to the power 0.25")[$( 3 x )^(0.25)$] + #math.equation(block: false, alt: "6 to the power minus 1 / 3")[$6^(− 1 / 3)$] + #math.equation(block: false, alt: "3 open parenthesis x y close parenthesis to the power minus 0.125")[$3 ( x y )^(− 0.125)$] + #math.equation(block: false, alt: "open parenthesis x minus 2 close parenthesis to the power 1 / 4")[$( x − 2 )^(1 / 4)$] + #math.equation(block: false, alt: "the fraction 1 over the cube root of 6")[$display(frac(1, root(3, 6)))$] + #math.equation(block: false, alt: "the fraction 3 over the 8th root of x y")[$display(frac(3, root(8, x y)))$] + #math.equation(block: false, alt: "the 4th root of x minus 2")[$root(4, x − 2)$] + #math.equation(block: false, alt: "8 to the power minus 1 / 4")[$8^(− 1 / 4)$] + #math.equation(block: false, alt: "y open parenthesis 5 x close parenthesis to the power minus 0.5")[$y ( 5 x )^(− 0.5)$] + #math.equation(block: false, alt: "open parenthesis y plus 2 close parenthesis to the power 1 / 3")[$( y + 2 )^(1 / 3)$] For Problems 13–16, write each expression in exponential form. + #math.equation(block: false, alt: "the square root of 7")[$sqrt(7)$] + #math.equation(block: false, alt: "the cube root of 2 x")[$root(3, 2 x)$] + #math.equation(block: false, alt: "2 the 5th root of z")[$2 root(5, z)$] + #math.equation(block: false, alt: "7 to the power 1 / 2")[$7^(1 / 2)$] + #math.equation(block: false, alt: "open parenthesis 2 x close parenthesis to the power 1 / 3")[$( 2 x )^(1 / 3)$] + #math.equation(block: false, alt: "2 z to the power 1 / 5")[$2 z^(1 / 5)$] + #math.equation(block: false, alt: "the square root of 5")[$sqrt(5)$] + #math.equation(block: false, alt: "the cube root of 4 y")[$root(3, 4 y)$] + #math.equation(block: false, alt: "5 the cube root of x")[$5 root(3, x)$] + #math.equation(block: false, alt: "the fraction minus 3 over the 4th root of 6")[$display(frac(− 3, root(4, 6)))$] + #math.equation(block: false, alt: "the 4th root of x minus 3 y")[$root(4, x − 3 y)$] + #math.equation(block: false, alt: "the fraction minus 1 over the 5th root of 1 plus 3 b")[$display(frac(− 1, root(5, 1 + 3 b)))$] + #math.equation(block: false, alt: "minus 3 times 6 to the power minus 1 / 4")[$− 3 ⋅ 6^(− 1 / 4)$] + #math.equation(block: false, alt: "open parenthesis x minus 3 y close parenthesis to the power 1 / 4")[$( x − 3 y )^(1 / 4)$] + #math.equation(block: false, alt: "minus open parenthesis 1 plus 3 b close parenthesis to the power minus 1 / 5")[$− ( 1 + 3 b )^(− 1 / 5)$] + #math.equation(block: false, alt: "the fraction 2 over the 5th root of 3")[$display(frac(2, root(5, 3)))$] + #math.equation(block: false, alt: "the cube root of y plus 2 x")[$root(3, y + 2 x)$] + #math.equation(block: false, alt: "the fraction minus 1 over the 4th root of 3 a minus 2 b")[$display(frac(− 1, root(4, 3 a − 2 b)))$] For Problems 17–18, simplify. + #math.equation(block: false, alt: "open parenthesis the cube root of 125 close parenthesis cubed")[$attach(( root(3, 125) ), t: 3)$] + #math.equation(block: false, alt: "open parenthesis the 4th root of 2 close parenthesis to the power 4")[$attach(( root(4, 2) ), t: 4)$] + #math.equation(block: false, alt: "open parenthesis 3 the square root of 7 close parenthesis squared")[$attach(( 3 sqrt(7) ), t: 2)$] + #math.equation(block: false, alt: "open parenthesis minus x squared the cube root of 2 x close parenthesis cubed")[$attach(( − x^(2) root(3, 2 x) ), t: 3)$] + #math.equation(block: false, alt: "125")[$125$] + #math.equation(block: false, alt: "2")[$2$] + #math.equation(block: false, alt: "63")[$63$] + #math.equation(block: false, alt: "minus 2 x to the power 7")[$− 2 x^(7)$] + #math.equation(block: false, alt: "open parenthesis the 4th root of 16 close parenthesis to the power 4")[$attach(( root(4, 16) ), t: 4)$] + #math.equation(block: false, alt: "open parenthesis the cube root of 6 close parenthesis cubed")[$attach(( root(3, 6) ), t: 3)$] + #math.equation(block: false, alt: "open parenthesis 2 the cube root of 12 close parenthesis cubed")[$attach(( 2 root(3, 12) ), t: 3)$] + #math.equation(block: false, alt: "open parenthesis minus a cubed the 4th root of a squared close parenthesis to the power 4")[$attach(( − a^(3) root(4, a^(2)) ), t: 4)$] For Problems 19–20, use a calculator to approximate each irrational number to the nearest thousandth. + #math.equation(block: false, alt: "2 to the power 1 / 2")[$2^(1 / 2)$] + #math.equation(block: false, alt: "the cube root of 75")[$root(3, 75)$] + #math.equation(block: false, alt: "the 4th root of 1.6")[$root(4, 1.6)$] + #math.equation(block: false, alt: "365 to the power minus 1 / 3")[$365^(− 1 / 3)$] + #math.equation(block: false, alt: "0.006 to the power minus 0.2")[$0.006^(− 0.2)$] + #math.equation(block: false, alt: "1.414")[$1.414$] + #math.equation(block: false, alt: "4.217")[$4.217$] + #math.equation(block: false, alt: "1.125")[$1.125$] + #math.equation(block: false, alt: "0.140")[$0.140$] + #math.equation(block: false, alt: "2.782")[$2.782$] + #math.equation(block: false, alt: "3 to the power 1 / 2")[$3^(1 / 2)$] + #math.equation(block: false, alt: "the 4th root of 60")[$root(4, 60)$] + #math.equation(block: false, alt: "the cube root of 1.4")[$root(3, 1.4)$] + #math.equation(block: false, alt: "1058 to the power minus 1 / 5")[$1058^(− 1 / 5)$] + #math.equation(block: false, alt: "1.05 to the power minus 0.1")[$1.05^(− 0.1)$] For Problems 21–22, write each expression as a power function. + #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals 3.7 the cube root of x")[$g ( x ) = 3.7 root(3, x)$] + #math.equation(block: false, alt: "H open parenthesis x close parenthesis equals the 4th root of 85 x")[$H ( x ) = root(4, 85 x)$] + #math.equation(block: false, alt: "F open parenthesis t close parenthesis equals the fraction 25 over the 5th root of t")[$F ( t ) = display(frac(25, root(5, t)))$] + #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals 3.7 x to the power 1 / 3")[$g ( x ) = 3.7 x^(1 / 3)$] + #math.equation(block: false, alt: "H open parenthesis x close parenthesis equals 85 to the power 1 / 4 x to the power 1 / 4")[$H ( x ) = 85^(1 / 4) x^(1 / 4)$] + #math.equation(block: false, alt: "F open parenthesis t close parenthesis equals 25 t to the power minus 1 / 5")[$F ( t ) = 25 t^(− 1 / 5)$] + #math.equation(block: false, alt: "h open parenthesis v close parenthesis equals 12.7 the square root of v")[$h ( v ) = 12.7 sqrt(v)$] + #math.equation(block: false, alt: "F open parenthesis p close parenthesis equals the cube root of 2.9 p")[$F ( p ) = root(3, 2.9 p)$] + #math.equation(block: false, alt: "G open parenthesis w close parenthesis equals the fraction 5 over 8 the 8th root of w")[$G ( w ) = display(frac(5, 8 root(8, w)))$] For Problems 23–30, solve. #math.equation(block: true, alt: "6.5 x to the power 1 / 3 plus 3.8 equals 33.05")[$6.5 x^(1 / 3) + 3.8 = 33.05$] #math.equation(block: true, alt: "x equals 91.125")[$x = 91.125$] #math.equation(block: true, alt: "9.8 minus 76 x to the power 1 / 4 plus 15 equals 9.6")[$9.8 − 76 x^(1 / 4) + 15 = 9.6$] #math.equation(block: true, alt: "4 open parenthesis x plus 2 close parenthesis to the power 1 / 5 equals 12")[$4 ( x + 2 )^(1 / 5) = 12$] #math.equation(block: true, alt: "x equals 241")[$x = 241$] #math.equation(block: true, alt: "minus 9 open parenthesis x minus 3 close parenthesis to the power 1 / 5 equals 18")[$− 9 ( x − 3 )^(1 / 5) = 18$] #math.equation(block: true, alt: "open parenthesis 2 x minus 3 close parenthesis to the power minus 1 / 4 equals the fraction 1 over 2")[$( 2 x − 3 )^(− 1 / 4) = display(frac(1, 2))$] #math.equation(block: true, alt: "x equals the fraction 19 over 2")[$x = display(frac(19, 2))$] #math.equation(block: true, alt: "open parenthesis 5 x plus 2 close parenthesis to the power minus 1 / 3 equals the fraction 1 over 4")[$( 5 x + 2 )^(− 1 / 3) = display(frac(1, 4))$] #math.equation(block: true, alt: "the cube root of x squared minus 3 equals 3")[$root(3, x^(2) − 3) = 3$] #math.equation(block: true, alt: "x equals plus or minus the square root of 30")[$x = ± sqrt(30)$] #math.equation(block: true, alt: "the 4th root of x cubed minus 7 equals 2")[$root(4, x^(3) − 7) = 2$] For Problems 31–38, solve for the indicated variable. #math.equation(block: false, alt: "T equals 2 π the square root of the fraction L over g")[$T = 2 π sqrt(display(frac(L, g))) " " " " " " " "$] for #math.equation(block: false, alt: "L")[$L$] #math.equation(block: true, alt: "L equals the fraction g T squared over 4 π squared")[$L = display(frac(g T^(2), 4 π^(2)))$] #math.equation(block: false, alt: "T equals 2 π the square root of the fraction m over k")[$T = 2 π sqrt(display(frac(m, k))) " " " " " " " "$] for #math.equation(block: false, alt: "m")[$m$] #math.equation(block: false, alt: "r equals the square root of t squared minus s squared")[$r = sqrt(t^(2) − s^(2)) " " " " " " " "$] for #math.equation(block: false, alt: "s")[$s$] #math.equation(block: true, alt: "s equals plus or minus the square root of t squared minus r squared")[$s = ± sqrt(t^(2) − r^(2))$] #math.equation(block: false, alt: "c equals the square root of a squared minus b squared")[$c = sqrt(a^(2) − b^(2)) " " " " " " " "$] for #math.equation(block: false, alt: "b")[$b$] #math.equation(block: false, alt: "r equals the cube root of the fraction 3 V over 4 π")[$r = root(3, display(frac(3 V, 4 π))) " " " " " " " "$] for #math.equation(block: false, alt: "V")[$V$] #math.equation(block: true, alt: "v equals the fraction 4 over 3 π r cubed")[$v = display(frac(4, 3)) π r^(3)$] #math.equation(block: false, alt: "d equals the cube root of the fraction 16 M r squared over m")[$d = root(3, display(frac(16 M r^(2), m))) " " " " " " " "$] for #math.equation(block: false, alt: "M")[$M$] #math.equation(block: false, alt: "R equals the 4th root of the fraction 8 L v f over π p")[$R = root(4, display(frac(8 L v f, π p))) " " " " " " " "$] for #math.equation(block: false, alt: "p")[$p$] #math.equation(block: true, alt: "p equals the fraction 8 L v f over π R to the power 4")[$p = display(frac(8 L v f, π R^(4)))$] #math.equation(block: false, alt: "T equals the 4th root of the fraction E over S A")[$T = root(4, display(frac(E, S A))) " " " " " " " "$] for #math.equation(block: false, alt: "A")[$A$] The period of a pendulum is the time it takes for the pendulum to complete one entire swing, from left to right and back again. The greater the length, #math.equation(block: false, alt: "L")[$L$], of the pendulum, the longer its period, #math.equation(block: false, alt: "T")[$T$]. In fact, if #math.equation(block: false, alt: "L")[$L$] is measured in feet, then the period is given in seconds by #math.equation(block: true, alt: "T equals 2 π the square root of the fraction L over 32")[$T = 2 π sqrt(frac(L, 32))$] + Write the formula for #math.equation(block: false, alt: "T")[$T$] as a power function in the form #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals k x to the power p")[$f ( x ) = k x^(p)$]. + Suppose you are standing in the Convention Center in Portland, Oregon, and you time the period of its Foucault pendulum (the longest in the world). Its period is approximately #math.equation(block: false, alt: "10.54")[$10.54$] seconds. How long is the pendulum? + Choose a reasonable domain for the function #math.equation(block: false, alt: "T equals f open parenthesis L close parenthesis")[$T = f ( L )$] and graph the function. + #math.equation(block: false, alt: "T equals the fraction 2 π over the square root of 32 L to the power 1 / 2")[$T = display(frac(2 π, sqrt(32))) L^(1 / 2)$] + #math.equation(block: false, alt: "90")[$90$] feet + #figure(figph[square root], alt: "square root", caption: none) If you are flying in an airplane at an altitude of #math.equation(block: false, alt: "h")[$h$] miles, on a clear day you can see a distance of #math.equation(block: false, alt: "d")[$d$] miles to the horizon, where #math.equation(block: true, alt: "d equals the square root of 7920 h")[$d = sqrt(7920 h)$] + Write the formula for #math.equation(block: false, alt: "d")[$d$] as a power function in the form #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals k x to the power p")[$f ( x ) = k x^(p)$]. + Choose a reasonable domain for the function #math.equation(block: false, alt: "d equals f open parenthesis h close parenthesis")[$d = f ( h )$] and graph the function. + At what altitude will you be able to see for a distance of #math.equation(block: false, alt: "100")[$100$] miles? How high is that in feet? If you walk in the normal way, your maximum speed, #math.equation(block: false, alt: "v")[$v$], in meters per second, is limited by the length of your legs, #math.equation(block: false, alt: "r")[$r$], according to the formula #math.equation(block: true, alt: "v equals the square root of g r")[$v = sqrt(g r)$] where the constant #math.equation(block: false, alt: "g")[$g$] is approximately #math.equation(block: false, alt: "10")[$10$] meters per second squared. (Source: Alexander, 1992) + A typical adult man has legs about #math.equation(block: false, alt: "0.9")[$0.9$] meter long. How fast can he walk? + A typical four-year-old has legs #math.equation(block: false, alt: "0.5")[$0.5$] meter long. How fast can she walk? + Graph maximum walking speed as a function of leg length. + Race-walkers can walk as fast as #math.equation(block: false, alt: "4.4")[$4.4$] meters per second by rotating their hips so that the effective length of their legs is increased. What is that effective length? + On the Moon the value of #math.equation(block: false, alt: "g")[$g$] is #math.equation(block: false, alt: "1.6")[$1.6$] meters per second squared. How fast can a typical adult man walk on the Moon? + #math.equation(block: false, alt: "3")[$3$] meters per second + #math.equation(block: false, alt: "approximately equals 2.2")[$≈ 2.2$] meters per second + #figure(figph[square root], alt: "square root", caption: none) + #math.equation(block: false, alt: "1.9")[$1.9$] meters + #math.equation(block: false, alt: "1.2")[$1.2$] meters per second When a ship moves through the water, it creates waves that impede its own progress. Because of this resistance, there is an upper limit to the speed at which a ship can travel, given, in knots, by #math.equation(block: true, alt: "v sub max equals 1.3 the square root of L")[$v_("max") = 1.3 sqrt(L)$] where #math.equation(block: false, alt: "L")[$L$] is the length of the vessel, in feet. (Source: Gilner, 1972) + Graph maximum speed as a function of vessel length. + The world's largest ship, the oil tanker #emph[Jahre Viking], is #math.equation(block: false, alt: "1054")[$1054$] feet long. What is its top speed? + As a ship approaches its maximum speed, the power required increases sharply. Therefore, most merchant ships are designed to cruise at speeds no higher than #math.equation(block: false, alt: "v sub c equals 0.8 the square root of L")[$v_(c) = 0.8 sqrt(L)$]. Graph #math.equation(block: false, alt: "v sub c")[$v_(c)$] on the same axes with #math.equation(block: false, alt: "v sub max")[$v_("max")$]. + What is the cruising speed of the #emph[Jahre Viking]? What percent of its maximum speed is that? A rough estimate for the radius of the nucleus of an atom is provided by the formula #math.equation(block: true, alt: "r equals k A to the power 1 / 3")[$r = k A^(1 / 3)$] where #math.equation(block: false, alt: "A")[$A$] is the mass number of the nucleus and #math.equation(block: false, alt: "k approximately equals 1.3 times 10 to the power minus 13")[$k ≈ 1.3 × 10^(− 13)$] centimeter. + Estimate the radius of the nucleus of an atom of iodine-127, which has mass number #math.equation(block: false, alt: "127")[$127$]. If the nucleus is roughly spherical, what is its volume? + The nuclear mass of iodine-127 is #math.equation(block: false, alt: "2.1 times 10 to the power minus 22")[$2.1 × 10^(− 22)$] gram. What is the density of the nucleus? (Density is mass per unit volume.) + Complete the table of values for the radii of various radioisotopes.#figure(table( columns: 6, align: left, inset: 6pt, table.header([Element], [Carbon], [Potassium], [Cobalt], [Technetium], [Radium]), [Mass #linebreak() number, #math.equation(block: false, alt: "A")[$A$] #linebreak()], [#math.equation(block: false, alt: "14")[$14$]], [#math.equation(block: false, alt: "40")[$40$]], [#math.equation(block: false, alt: "60")[$60$]], [#math.equation(block: false, alt: "99")[$99$]], [#math.equation(block: false, alt: "226")[$226$]], [Radius, #math.equation(block: false, alt: "r")[$r$]], [], [], [], [], [], )) + Sketch a graph of #math.equation(block: false, alt: "r")[$r$] as a function of #math.equation(block: false, alt: "A")[$A$]. (Use units of #math.equation(block: false, alt: "10 to the power minus 13")[$10^(− 13)$] centimeter on the vertical axis.) + #math.equation(block: false, alt: "6.5 times 10 to the power minus 13")[$6.5 × 10^(− 13)$] cm; #math.equation(block: false, alt: "1.17 times 10 to the power minus 36 cm cubed")[$1.17 × 10^(− 36) attach(" cm", t: 3)$] + #math.equation(block: false, alt: "1.8 times 10 to the power 14 g / cm cubed")[$1.8 × 10^(14) g / "cm"^(3)$] + #figure(table( columns: 6, align: left, inset: 6pt, table.header([Element], [Carbon], [Potassium], [Cobalt], [Technetium], [Radium]), [Mass #linebreak() number, #math.equation(block: false, alt: "A")[$A$] #linebreak()], [#math.equation(block: false, alt: "14")[$14$]], [#math.equation(block: false, alt: "40")[$40$]], [#math.equation(block: false, alt: "60")[$60$]], [#math.equation(block: false, alt: "99")[$99$]], [#math.equation(block: false, alt: "226")[$226$]], [Radius, #math.equation(block: false, alt: "r")[$r$] #linebreak() (#math.equation(block: false, alt: "10 to the power minus 13")[$10^(− 13)$] cm) #linebreak()], [#math.equation(block: false, alt: "3.1")[$3.1$]], [#math.equation(block: false, alt: "4.4")[$4.4$]], [#math.equation(block: false, alt: "5.1")[$5.1$]], [#math.equation(block: false, alt: "6")[$6$]], [#math.equation(block: false, alt: "7.9")[$7.9$]], )) + #figure(figph[cube root], alt: "cube root", caption: none) In the sport of men's crew racing, the best times vary closely with the number of men in the crew, according to the formula #math.equation(block: true, alt: "t equals k n to the power minus 1 / 9")[$t = k n^(− 1 / 9)$] where #math.equation(block: false, alt: "n")[$n$] is the number of men in the crew and #math.equation(block: false, alt: "t")[$t$] is the winning time, in minutes, for a #math.equation(block: false, alt: "2000")[$2000$]-meter race. + If the winning time for the #math.equation(block: false, alt: "8")[$8$]-man crew was #math.equation(block: false, alt: "5.73")[$5.73$] minutes, estimate the value of #math.equation(block: false, alt: "k")[$k$]. + Complete the table of values of predicted winning times for the other racing classes.#figure(table( columns: 5, align: left, inset: 6pt, table.header([Size of crew, #math.equation(block: false, alt: "n")[$n$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "8")[$8$]]), [Winning time, #math.equation(block: false, alt: "t")[$t$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], )) + Sketch a graph of #math.equation(block: false, alt: "t")[$t$] as a function of #math.equation(block: false, alt: "n")[$n$]. In Problems 45–48, one quantity varies directly with the square root of the other, that is, #math.equation(block: false, alt: "y equals k the square root of x")[$y = k sqrt(x)$]. + Find the value of #math.equation(block: false, alt: "k")[$k$] and write a power function relating the variables. + Use your function to answer the question. + Graph your function and verify your answer to part (b) on the graph. The stream speed necessary to move a granite particle is a function of the diameter of the particle; faster river currents can move larger particles. The table shows the stream speed necessary to move particles of different sizes. What speed is needed to carry a particle with diameter #math.equation(block: false, alt: "0.36")[$0.36$] centimeter? #figure(table( columns: 2, align: left, inset: 6pt, table.header([Diameter, #math.equation(block: false, alt: "d")[$d$] (cm)], [Speed, #math.equation(block: false, alt: "s")[$s$] (cm/sec)]), [#math.equation(block: false, alt: "0.01")[$0.01$]], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "0.04")[$0.04$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "0.09")[$0.09$]], [#math.equation(block: false, alt: "15")[$15$]], [#math.equation(block: false, alt: "0.16")[$0.16$]], [#math.equation(block: false, alt: "20")[$20$]], )) + #math.equation(block: false, alt: "s equals 50 the square root of d")[$s = 50 sqrt(d)$] + #math.equation(block: false, alt: "30")[$30$] cm/sec + #figure(figph[square root], alt: "square root", caption: none) The speed at which water comes out of the spigot at the bottom of a water jug is a function of the water level in the jug; it slows down as the water level drops. The table shows different water levels and the resulting flow speeds. What is the flow speed when the water level is at #math.equation(block: false, alt: "16")[$16$] inches? #figure(table( columns: 2, align: left, inset: 6pt, table.header([Level, #math.equation(block: false, alt: "L")[$L$] (in)], [Speed, #math.equation(block: false, alt: "s")[$s$] (gal/min)]), [#math.equation(block: false, alt: "9")[$9$]], [#math.equation(block: false, alt: "1.5")[$1.5$]], [#math.equation(block: false, alt: "6.25")[$6.25$]], [#math.equation(block: false, alt: "1.25")[$1.25$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2.25")[$2.25$]], [#math.equation(block: false, alt: "0.75")[$0.75$]], )) The rate, #math.equation(block: false, alt: "r")[$r$], in feet per second, at which water flows from a fire hose is a function of the water pressure, #math.equation(block: false, alt: "P")[$P$], in psi (pounds per square inch). What is the rate of water flow at a typical water pressure of #math.equation(block: false, alt: "60")[$60$] psi? #figure(table( columns: 5, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "P")[$P$] (psi)], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "30")[$30$]], [#math.equation(block: false, alt: "40")[$40$]]), [#math.equation(block: false, alt: "r")[$r$] (ft/sec)], [#math.equation(block: false, alt: "38.3")[$38.3$]], [#math.equation(block: false, alt: "54.1")[$54.1$]], [#math.equation(block: false, alt: "66.3")[$66.3$]], [#math.equation(block: false, alt: "76.5")[$76.5$]], )) + #math.equation(block: false, alt: "r equals 12.1 the square root of P")[$r = 12.1 sqrt(P)$] + #math.equation(block: false, alt: "94")[$94$] ft/sec + #figure(figph[square root], alt: "square root", caption: none) When a layer of ice forms on a pond, the thickness of the ice, #math.equation(block: false, alt: "d")[$d$], in centimeters, is a function of time, #math.equation(block: false, alt: "t")[$t$], in minutes. How thick is the ice after #math.equation(block: false, alt: "3")[$3$] hours? #figure(table( columns: 5, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "t")[$t$] (min)], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "30")[$30$]], [#math.equation(block: false, alt: "40")[$40$]], [#math.equation(block: false, alt: "60")[$60$]]), [#math.equation(block: false, alt: "d")[$d$] (cm)], [#math.equation(block: false, alt: "0.50")[$0.50$]], [#math.equation(block: false, alt: "0.87")[$0.87$]], [#math.equation(block: false, alt: "1.01")[$1.01$]], [#math.equation(block: false, alt: "1.24")[$1.24$]], )) Membership in the County Museum has been increasing since it was built in #math.equation(block: false, alt: "1980")[$1980$]. The number of members is given by the function #math.equation(block: true, alt: "M open parenthesis t close parenthesis equals 72 plus 100 t to the power 1 / 3")[$M ( t ) = 72 + 100 t^(1 / 3)$] where #math.equation(block: false, alt: "t")[$t$] is the number of years since #math.equation(block: false, alt: "1980")[$1980$]. + How many members were there in #math.equation(block: false, alt: "1990")[$1990$]? In #math.equation(block: false, alt: "2000")[$2000$]? + In what year will the museum have #math.equation(block: false, alt: "400")[$400$] members? If the membership continues to grow according to the given function, when will the museum have #math.equation(block: false, alt: "500")[$500$] members? + Graph the function #math.equation(block: false, alt: "M open parenthesis t close parenthesis")[$M ( t )$]. How would you describe the growth of the membership over time? + #math.equation(block: false, alt: "287")[$287$]; #math.equation(block: false, alt: "343")[$343$] + #math.equation(block: false, alt: "2015")[$2015$]; #math.equation(block: false, alt: "2058")[$2058$] + The membership grows rapidly at first but is growing less rapidly with time.#figure(figph[cube root], alt: "cube root", caption: none) Due to improvements in technology, the annual electricity cost of running most major appliances has decreased steadily since #math.equation(block: false, alt: "1970")[$1970$]. The average annual cost of running a refrigerator is given, in dollars, by the function #math.equation(block: true, alt: "C open parenthesis t close parenthesis equals 148 minus 28 t to the power 1 / 3")[$C ( t ) = 148 − 28 t^(1 / 3)$] where #math.equation(block: false, alt: "t")[$t$] is the number of years since #math.equation(block: false, alt: "1970")[$1970$]. + How much did it cost to run a refrigerator in #math.equation(block: false, alt: "1980")[$1980$]? In #math.equation(block: false, alt: "1990")[$1990$]? + When was the cost of running a refrigerator half of the cost in #math.equation(block: false, alt: "1970")[$1970$]? If the cost continues to decline according to the given function, when will it cost \$#math.equation(block: false, alt: "50")[$50$] per year to run a refrigerator? + Graph the function #math.equation(block: false, alt: "C open parenthesis t close parenthesis")[$C ( t )$]. Do you think that the cost will continue to decline indefinitely according to the given function? Why or why not? Match each function with the description of its graph in the first quadrant. + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x squared")[$f ( x ) = x^(2)$] + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x to the power minus 2")[$f ( x ) = x^(− 2)$] + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x to the power 1 / 2")[$f ( x ) = x^(1 / 2)$] + Increasing and concave up + Increasing and concave down + Decreasing and concave up + Decreasing and concave down + I + III + II + none In each pair, match the functions with their graphs. #figure(figph[three pairs of curves], alt: "three pairs of curves", caption: none) + Graph the functions #math.equation(block: true, alt: "y sub 1 equals x to the power 1 / 2 , y sub 2 equals x to the power 1 / 3 , y sub 3 equals x to the power 1 / 4 , y sub 4 equals x to the power 1 / 5")[$y_(1) = x^(1 / 2) , " " " " " " y_(2) = x^(1 / 3) , " " " " " " y_(3) = x^(1 / 4) , " " " " " " y_(4) = x^(1 / 5)$] in the window #math.equation(block: true, alt: "Xmin, equals 0, Xmax equals 100; Ymin, equals 0, Ymax equals 10")[$"Xmin" & = 0 & & "Xmax" = 100 \ "Ymin" & = 0 & & "Ymax" = 10$] What do you observe? + Use your graphs to evaluate #math.equation(block: false, alt: "100 to the power 1 / 2")[$100^(1 / 2)$], #math.equation(block: false, alt: "100 to the power 1 / 3")[$100^(1 / 3)$], #math.equation(block: false, alt: "100 to the power 1 / 4")[$100^(1 / 4)$], and #math.equation(block: false, alt: "100 to the power 1 / 5")[$100^(1 / 5)$]. + Use your calculator to evaluate #math.equation(block: false, alt: "100 to the power 1 / n")[$100^(1 / n)$] for #math.equation(block: false, alt: "n equals 10")[$n = 10$], #math.equation(block: false, alt: "n equals 100")[$n = 100$], and #math.equation(block: false, alt: "n equals 1000")[$n = 1000$]. What happens when #math.equation(block: false, alt: "n")[$n$] gets large? + The graphs of #math.equation(block: false, alt: "x to the power 1 / n")[$x^(1 / n)$] become closer and closer to horizontal when #math.equation(block: false, alt: "n")[$n$] increases (for #math.equation(block: false, alt: "x greater than 1")[$x > 1$]). + #math.equation(block: false, alt: "10 , 4.64 , 3.16 , 2.51")[$10 , " " 4.64 , " " 3.16 , " " 2.51$] + #math.equation(block: false, alt: "1.58 , 1.05 , 1.005")[$1.58 , 1.05 , " " 1.005$]; the values decrease towards #math.equation(block: false, alt: "1")[$1$]. + Graph the functions #math.equation(block: true, alt: "y sub 1 equals x to the power 1 / 2 , y sub 2 equals x to the power 1 / 3 , y sub 3 equals x to the power 1 / 4 , y sub 4 equals x to the power 1 / 5")[$y_(1) = x^(1 / 2) , " " " " " " y_(2) = x^(1 / 3) , " " " " " " y_(3) = x^(1 / 4) , " " " " " " y_(4) = x^(1 / 5)$] in the window #math.equation(block: true, alt: "Xmin, equals 0, Xmax equals 1; Ymin, equals 0, Ymax equals 1")[$"Xmin" & = 0 & & "Xmax" = 1 \ "Ymin" & = 0 & & "Ymax" = 1$] What do you observe? + Use your graphs to evaluate #math.equation(block: false, alt: "0.5 to the power 1 / 2")[$0.5^(1 / 2)$], #math.equation(block: false, alt: "0.5 to the power 1 / 3")[$0.5^(1 / 3)$], #math.equation(block: false, alt: "0.5 to the power 1 / 4")[$0.5^(1 / 4)$], and #math.equation(block: false, alt: "0.5 to the power 1 / 5")[$0.5^(1 / 5)$]. + Use your calculator to evaluate #math.equation(block: false, alt: "0.5 to the power 1 / n")[$0.5^(1 / n)$] for #math.equation(block: false, alt: "n equals 10")[$n = 10$], #math.equation(block: false, alt: "n equals 100")[$n = 100$], and #math.equation(block: false, alt: "n equals 1000")[$n = 1000$]. What happens when #math.equation(block: false, alt: "n")[$n$] gets large? For Problems 55–58, graph each set of functions in the given window. What do you observe? #math.equation(block: true, alt: "y sub 1 equals the square root of x , y sub 2 equals x squared , y sub 3 equals x")[$y_(1) = sqrt(x) , " " " " " " y_(2) = x^(2) , " " " " " " y_(3) = x$] #math.equation(block: true, alt: "Xmin, equals 0, Xmax equals 4; Ymin, equals 0, Ymax equals 4")[$"Xmin" & = 0 & & "Xmax" = 4 \ "Ymin" & = 0 & & "Ymax" = 4$] The graphs of #math.equation(block: false, alt: "y sub 1")[$y_(1)$] and #math.equation(block: false, alt: "y sub 2")[$y_(2)$] are symmetric about #math.equation(block: false, alt: "y sub 3 equals x")[$y_(3) = x$]. #math.equation(block: true, alt: "y sub 1 equals the cube root of x , y sub 2 equals x cubed , y sub 3 equals x")[$y_(1) = root(3, x) , " " " " " " y_(2) = x^(3) , " " " " " " y_(3) = x$] #math.equation(block: true, alt: "Xmin, equals minus 4, Xmax equals 4; Ymin, equals minus 4, Ymax equals 4")[$"Xmin" & = − 4 & & "Xmax" = 4 \ "Ymin" & = − 4 & & "Ymax" = 4$] #math.equation(block: true, alt: "y sub 1 equals the 5th root of x , y sub 2 equals x to the power 5 , y sub 3 equals x")[$y_(1) = root(5, x) , " " " " " " y_(2) = x^(5) , " " " " " " y_(3) = x$] #math.equation(block: true, alt: "Xmin, equals minus 2, Xmax equals 2; Ymin, equals minus 2, Ymax equals 2")[$"Xmin" & = − 2 & & "Xmax" = 2 \ "Ymin" & = − 2 & & "Ymax" = 2$] The graphs of #math.equation(block: false, alt: "y sub 1")[$y_(1)$] and #math.equation(block: false, alt: "y sub 2")[$y_(2)$] are symmetric about #math.equation(block: false, alt: "y sub 3 equals x")[$y_(3) = x$]. #math.equation(block: true, alt: "y sub 1 equals the 4th root of x , y sub 2 equals x to the power 4 , y sub 3 equals x")[$y_(1) = root(4, x) , " " " " " " y_(2) = x^(4) , " " " " " " y_(3) = x$] #math.equation(block: true, alt: "Xmin, equals 0, Xmax equals 2; Ymin, equals 0, Ymax equals 2")[$"Xmin" & = 0 & & "Xmax" = 2 \ "Ymin" & = 0 & & "Ymax" = 2$] + Graph the functions #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 4 the cube root of x minus 9")[$" " " " f ( x ) = 4 root(3, x − 9) " " " "$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals 12")[$" " " " g ( x ) = 12 " " " "$] in the window #math.equation(block: true, alt: "Xmin, equals 0, Xmax equals 47; Ymin, equals minus 8, Ymax equals 16")[$"Xmin" & = 0 & & "Xmax" = 47 \ "Ymin" & = − 8 & & "Ymax" = 16$] + Use the graph to solve the equation #math.equation(block: false, alt: "4 the cube root of x minus 9 equals 12")[$4 root(3, x − 9) = 12$]. + Solve the equation algebraically. + #figure(figph[cube root], alt: "cube root", caption: none) + #math.equation(block: false, alt: "x equals 36")[$x = 36$] + Graph the functions #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 6 plus 2 the 4th root of 12 minus x")[$" " " " f ( x ) = 6 + 2 root(4, 12 − x) " " " "$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals 10")[$" " " " g ( x ) = 10 " " " "$] in the window #math.equation(block: true, alt: "Xmin, equals minus 27, Xmax equals 20; Ymin, equals 4, Ymax equals 12")[$"Xmin" & = − 27 & & "Xmax" = 20 \ "Ymin" & = 4 & & "Ymax" = 12$] + Use the graph to solve the equation #math.equation(block: false, alt: "6 plus 2 the 4th root of 12 minus x equals 10")[$6 + 2 root(4, 12 − x) = 10$]. + Solve the equation algebraically. + Write #math.equation(block: false, alt: "the square root of x")[$sqrt(x)$] with a fractional exponent. + Write #math.equation(block: false, alt: "the square root of the square root of x")[$display(sqrt(sqrt(x)))$] with fractional exponents. + Use the laws of exponents to show that #math.equation(block: false, alt: "the square root of the square root of x equals the 4th root of x")[$display(sqrt(sqrt(x)) = root(4, x))$]. + #math.equation(block: false, alt: "x to the power 1 / 2")[$x^(1 / 2)$] + #math.equation(block: false, alt: "open parenthesis x to the power 1 / 2 close parenthesis to the power 1 / 2")[$attach(( x^(1 / 2) ), t: 1 / 2)$] + #math.equation(block: true, alt: "the square root of the square root of x, equals open parenthesis x to the power 1 / 2 close parenthesis to the power 1 / 2, By definition of fractional exponents.; equals x to the power 1 / 4, By the third law of exponents.; equals the 4th root of x, By definition of fractional exponents.")[$sqrt(sqrt(x)) & = attach(( x^(1 / 2) ), t: 1 / 2) & & "By definition of fractional exponents." \ & = x^(1 / 4) & & "By the third law of exponents." \ & = root(4, x) & & "By definition of fractional exponents."$] + Write #math.equation(block: false, alt: "the cube root of x")[$root(3, x)$] with a fractional exponent. + Write #math.equation(block: false, alt: "the square root of the cube root of x")[$display(sqrt(root(3, x)))$] with a fractional exponents. + Use the laws of exponents to show that #math.equation(block: false, alt: "the square root of the cube root of x equals the 6th root of x")[$display(sqrt(root(3, x)) = root(6, x))$]. For Problems 63–68, write the expression as a sum of terms of the form #math.equation(block: false, alt: "k x to the power p")[$k x^(p)$]. #math.equation(block: true, alt: "the fraction the square root of x over 4 minus the fraction 2 over the square root of x plus the fraction x over the square root of 2")[$display(frac(sqrt(x), 4)) − display(frac(2, sqrt(x))) + display(frac(x, sqrt(2)))$] #math.equation(block: true, alt: "the fraction 1 over 4 x to the power 1 / 2 minus 2 x to the power minus 1 / 2 plus the fraction 1 over the square root of 2 x")[$display(frac(1, 4) x^(1 / 2) − 2 x^(− 1 / 2) + frac(1, sqrt(2)) x)$] #math.equation(block: true, alt: "the fraction the square root of 3 over x plus the fraction 3 over the square root of x minus the fraction the square root of x over 3")[$display(frac(sqrt(3), x)) + display(frac(3, sqrt(x))) − display(frac(sqrt(x), 3))$] #math.equation(block: true, alt: "the fraction 6 minus the cube root of x over 2 the cube root of x")[$display(frac(6 − root(3, x), 2 root(3, x)))$] #math.equation(block: true, alt: "3 x to the power minus 1 / 3 minus the fraction 1 over 2")[$display(3 x^(− 1 / 3) − frac(1, 2))$] #math.equation(block: true, alt: "the fraction the 4th root of x plus 2 over 2 the 4th root of x")[$display(frac(root(4, x) + 2, 2 root(4, x)))$] #math.equation(block: true, alt: "x to the power minus 0.5 open parenthesis x plus x to the power 0.25 minus x to the power 0.5 close parenthesis")[$x^(− 0.5) ( x + x^(0.25) − x^(0.5) )$] #math.equation(block: true, alt: "x to the power 0.5 plus x to the power minus 0.25 minus x to the power 0")[$x^(0.5) + x^(− 0.25) − x^(0)$] #math.equation(block: true, alt: "x to the power 0.5 open parenthesis x to the power minus 1 plus x to the power minus 0.5 plus x to the power minus 0.25 close parenthesis")[$x^(0.5) ( x^(− 1) + x^(− 0.5) + x^(− 0.25) )$]