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== 9.5#h(0.6em)Divide Square Roots

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Before you get started, take this readiness quiz.

Find a fraction equivalent to #math.equation(block: false, alt: "the fraction 5 over 8")[$frac(5, 8)$] with denominator 48. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "the fraction 30 over 48")[$frac(30, 48)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis the square root of 5 close parenthesis squared")[$attach(( sqrt(5) ), t: 2)$]. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "5")[$5$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "open parenthesis 7 plus 3 x close parenthesis open parenthesis 7 minus 3 x close parenthesis")[$( 7 + 3 x ) ( 7 − 3 x )$]. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "49 minus 9 x squared")[$49 − 9 x^(2)$]
]
]


=== Divide Square Roots

We know that we simplify fractions by removing factors common to the numerator and the denominator. When we have a fraction with a square root in the numerator, we first simplify the square root. Then we can look for common factors.

#figure(figph[This figure shows two columns. The first is labeled “Common Factors” and has 3 times the square root of 2 over 3 times 5 beneath it. Both number threes are red. The second column is labeled “No common factors” and has 2 times the square root of 3 over 3 times 5.], alt: "This figure shows two columns. The first is labeled “Common Factors” and has 3 times the square root of 2 over 3 times 5 beneath it. Both number threes are red. The second column is labeled “No common factors” and has 2 times the square root of 3 over 3 times 5.", caption: none)

#examplebox("Example 1")[][
Simplify: #math.equation(block: false, alt: "the fraction the square root of 54 over 6")[$frac(sqrt(54), 6)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction the square root of 54 over 6")[$frac(sqrt(54), 6)$]]),
  [Simplify the radical.], [#math.equation(block: false, alt: "the fraction the square root of 9 times the square root of 6 over 6")[$frac(sqrt(9) · sqrt(6), 6)$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 3 the square root of 6 over 6")[$frac(3 sqrt(6), 6)$]],
  [Remove the common factors.], [#math.equation(block: false, alt: "the fraction 3 the square root of 6 over 3 times 2")[$frac(cancel(3) sqrt(6), cancel(3) · 2)$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction the square root of 6 over 2")[$frac(sqrt(6), 2)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction the square root of 32 over 8")[$frac(sqrt(32), 8)$].

#solutionbox[
#math.equation(block: true, alt: "the fraction the square root of 2 over 2")[$frac(sqrt(2), 2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction the square root of 75 over 15")[$frac(sqrt(75), 15)$].

#solutionbox[
#math.equation(block: true, alt: "the fraction the square root of 3 over 3")[$frac(sqrt(3), 3)$]
]
]

#examplebox("Example 2")[][
Simplify: #math.equation(block: false, alt: "the fraction 6 minus the square root of 24 over 12")[$frac(6 − sqrt(24), 12)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 6 minus the square root of 24 over 12")[$frac(6 − sqrt(24), 12)$]]),
  [Simplify the radical.], [#math.equation(block: false, alt: "the fraction 6 minus the square root of 4 times the square root of 6 over 12")[$frac(6 − sqrt(4) · sqrt(6), 12)$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 6 minus 2 the square root of 6 over 12")[$frac(6 − 2 sqrt(6), 12)$]],
  [Factor the common factor from the numerator.], [#math.equation(block: false, alt: "the fraction 2 open parenthesis 3 minus the square root of 6 close parenthesis over 2 times 6")[$frac(2 ( 3 − sqrt(6) ), 2 · 6)$]],
  [Remove the common factors.], [#math.equation(block: false, alt: "the fraction 2 open parenthesis 3 minus the square root of 6 close parenthesis over 2 times 6")[$frac(cancel(2) ( 3 − sqrt(6) ), cancel(2) · 6)$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 3 minus the square root of 6 over 6")[$frac(3 − sqrt(6), 6)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 8 minus the square root of 40 over 10")[$frac(8 − sqrt(40), 10)$].

#solutionbox[
#math.equation(block: true, alt: "the fraction 4 minus the square root of 10 over 5")[$frac(4 − sqrt(10), 5)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 10 minus the square root of 75 over 20")[$frac(10 − sqrt(75), 20)$].

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 minus the square root of 3 over 4")[$frac(2 − sqrt(3), 4)$]
]
]

We have used the Quotient Property of Square Roots to simplify square roots of fractions. The Quotient Property of Square Roots says

#math.equation(block: true, alt: "the square root of the fraction a over b equals the fraction the square root of a over the square root of b , b not equal to 0")[$sqrt(frac(a, b)) = frac(sqrt(a), sqrt(b)) , b ≠ 0$]Sometimes we will need to use the Quotient Property of Square Roots ‘in reverse’ to simplify a fraction with square roots.

#math.equation(block: true, alt: "the fraction the square root of a over the square root of b equals the square root of the fraction a over b , b not equal to 0")[$frac(sqrt(a), sqrt(b)) = sqrt(frac(a, b)) , b ≠ 0$]We will rewrite the Quotient Property of Square Roots so we see both ways together. Remember: we assume all variables are greater than or equal to zero so that their square roots are real numbers.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Quotient Property of Square Roots]

If #emph[a], #emph[b] are non-negative real numbers and #math.equation(block: false, alt: "b not equal to 0")[$b ≠ 0$], then

#math.equation(block: true, alt: "the square root of the fraction a over b equals the fraction the square root of a over the square root of b and the fraction the square root of a over the square root of b equals the square root of the fraction a over b")[$sqrt(frac(a, b)) = frac(sqrt(a), sqrt(b)) #h(1em) "and" #h(1em) frac(sqrt(a), sqrt(b)) = sqrt(frac(a, b))$]
]

We will use the Quotient Property of Square Roots ‘in reverse’ when the fraction we start with is the quotient of two square roots, and neither radicand is a perfect square. When we write the fraction in a single square root, we may find common factors in the numerator and denominator.

#examplebox("Example 3")[][
Simplify: #math.equation(block: false, alt: "the fraction the square root of 27 over the square root of 75")[$frac(sqrt(27), sqrt(75))$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction the square root of 27 over the square root of 75")[$frac(sqrt(27), sqrt(75))$]]),
  [Neither radicand is a perfect square, so rewrite using the quotient property of square roots.], [#math.equation(block: false, alt: "the square root of the fraction 27 over 75")[$sqrt(frac(27, 75))$]],
  [Remove common factors in the numerator and denominator.], [#math.equation(block: false, alt: "the square root of the fraction 3 times 9 over 3 times 25")[$sqrt(frac(cancel(3) · 9, cancel(3) · 25))$]],
  [Simplify.], [#math.equation(block: false, alt: "the square root of the fraction 9 over 25")[$sqrt(frac(9, 25))$]],
  [], [#math.equation(block: false, alt: "the fraction 3 over 5")[$frac(3, 5)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction the square root of 48 over the square root of 108")[$frac(sqrt(48), sqrt(108))$].

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 over 3")[$frac(2, 3)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction the square root of 96 over the square root of 54")[$frac(sqrt(96), sqrt(54))$].

#solutionbox[
#math.equation(block: true, alt: "the fraction 4 over 3")[$frac(4, 3)$]
]
]

We will use the Quotient Property for Exponents, #math.equation(block: false, alt: "the fraction a to the power m over a to the power n equals a to the power m minus n")[$frac(a^(m), a^(n)) = a^(m − n)$], when we have variables with exponents in the radicands.

#examplebox("Example 4")[][
Simplify: #math.equation(block: false, alt: "the fraction the square root of 6 y to the power 5 over the square root of 2 y")[$frac(sqrt(6 y^(5)), sqrt(2 y))$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction the square root of 6 y to the power 5 over the square root of 2 y")[$frac(sqrt(6 y^(5)), sqrt(2 y))$]]),
  [Neither radicand is a perfect square, so rewrite using the quotient property of square roots.], [#math.equation(block: false, alt: "the square root of the fraction 6 y to the power 5 over 2 y")[$sqrt(frac(6 y^(5), 2 y))$]],
  [Remove common factors in the numerator and denominator.], [#math.equation(block: false, alt: "the square root of the fraction 2 times 3 times y to the power 4 times y over 2 times y")[$sqrt(frac(cancel(2) · 3 · y^(4) · cancel(y), cancel(2) · cancel(y)))$]],
  [Simplify.], [#math.equation(block: false, alt: "the square root of 3 y to the power 4")[$sqrt(3 y^(4))$]],
  [Simplify the radical.], [#math.equation(block: false, alt: "y squared the square root of 3")[$y^(2) sqrt(3)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction the square root of 12 r cubed over the square root of 6 r")[$frac(sqrt(12 r^(3)), sqrt(6 r))$].

#solutionbox[
#math.equation(block: true, alt: "r the square root of 2")[$r sqrt(2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction the square root of 14 p to the power 9 over the square root of 2 p to the power 5")[$frac(sqrt(14 p^(9)), sqrt(2 p^(5)))$].

#solutionbox[
#math.equation(block: true, alt: "p squared the square root of 7")[$p^(2) sqrt(7)$]
]
]

#examplebox("Example 5")[][
Simplify: #math.equation(block: false, alt: "the fraction the square root of 72 x cubed over the square root of 162 x")[$frac(sqrt(72 x^(3)), sqrt(162 x))$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction the square root of 72 x cubed over the square root of 162 x")[$frac(sqrt(72 x^(3)), sqrt(162 x))$]]),
  [Rewrite using the quotient property of square roots.], [#math.equation(block: false, alt: "the square root of the fraction 72 x cubed over 162 x")[$sqrt(frac(72 x^(3), 162 x))$]],
  [Remove common factors.], [#math.equation(block: false, alt: "the square root of the fraction 18 times 4 times x squared times x over 18 times 9 times x")[$sqrt(frac(cancel(18) · 4 · x^(2) · cancel(x), cancel(18) · 9 · cancel(x)))$]],
  [Simplify.], [#math.equation(block: false, alt: "the square root of the fraction 4 x squared over 9")[$sqrt(frac(4 x^(2), 9))$]],
  [Simplify the radical.], [#math.equation(block: false, alt: "the fraction 2 x over 3")[$frac(2 x, 3)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction the square root of 50 s cubed over the square root of 128 s")[$frac(sqrt(50 s^(3)), sqrt(128 s))$].

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 s over 8")[$frac(5 s, 8)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction the square root of 75 q to the power 5 over the square root of 108 q")[$frac(sqrt(75 q^(5)), sqrt(108 q))$].

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 q squared over 6")[$frac(5 q^(2), 6)$]
]
]

#examplebox("Example 6")[][
Simplify: #math.equation(block: false, alt: "the fraction the square root of 147 a b to the power 8 over the square root of 3 a cubed b to the power 4")[$frac(sqrt(147 a b^(8)), sqrt(3 a^(3) b^(4)))$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction the square root of 147 a b to the power 8 over the square root of 3 a cubed b to the power 4")[$frac(sqrt(147 a b^(8)), sqrt(3 a^(3) b^(4)))$]]),
  [Rewrite using the quotient property of square roots.], [#math.equation(block: false, alt: "the square root of the fraction 147 a b to the power 8 over 3 a cubed b to the power 4")[$sqrt(frac(147 a b^(8), 3 a^(3) b^(4)))$]],
  [Remove common factors.], [#math.equation(block: false, alt: "the square root of the fraction 49 b to the power 4 over a squared")[$sqrt(frac(49 b^(4), a^(2)))$]],
  [Simplify the radical.], [#math.equation(block: false, alt: "the fraction 7 b squared over a")[$frac(7 b^(2), a)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction the square root of 162 x to the power 10 y squared over the square root of 2 x to the power 6 y to the power 6")[$frac(sqrt(162 x^(10) y^(2)), sqrt(2 x^(6) y^(6)))$].

#solutionbox[
#math.equation(block: true, alt: "the fraction 9 x squared over y squared")[$frac(9 x^(2), y^(2))$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction the square root of 300 m cubed n to the power 7 over the square root of 3 m to the power 5 n")[$frac(sqrt(300 m^(3) n^(7)), sqrt(3 m^(5) n))$].

#solutionbox[
#math.equation(block: true, alt: "the fraction 10 n cubed over m")[$frac(10 n^(3), m)$]
]
]


=== Rationalize a One Term Denominator

Before the calculator became a tool of everyday life, tables of square roots were used to find approximate values of square roots.  shows a portion of a table of squares and square roots. Square roots are approximated to five decimal places in this table.

#figure(figph[This table has three solumn and eleven rows. The columns are labeled, “n,” “n squared,” and “the square root of n.” Under the column labeled “n” are the following numbers: 200; 201; 202; 203; 204; 205; 206; 207; 208; 209; and 210. Under the column labeled, “n squared” are the following numbers: 40,000; 40,401; 40,804; 41,209; 41,616; 42,025; 42,436; 42,849; 43,264; 43,681; 44,100. Under the column labeled, “the square root of n” are the following numbers: 14.14214; 14.17745; 14.21267; 14.24781; 14.28286; 14.31782; 14.35270; 14.38749; 14.42221; 14.45683; 14.49138.], alt: "This table has three solumn and eleven rows. The columns are labeled, “n,” “n squared,” and “the square root of n.” Under the column labeled “n” are the following numbers: 200; 201; 202; 203; 204; 205; 206; 207; 208; 209; and 210. Under the column labeled, “n squared” are the following numbers: 40,000; 40,401; 40,804; 41,209; 41,616; 42,025; 42,436; 42,849; 43,264; 43,681; 44,100. Under the column labeled, “the square root of n” are the following numbers: 14.14214; 14.17745; 14.21267; 14.24781; 14.28286; 14.31782; 14.35270; 14.38749; 14.42221; 14.45683; 14.49138.", caption: [A table of square roots was used to find approximate values of square roots before there were calculators.])

If someone needed to approximate a fraction with a square root in the denominator, it meant doing long division with a five decimal-place divisor. This was a very cumbersome process.

For this reason, a process called rationalizing the denominator was developed. A fraction with a radical in the denominator is converted to an equivalent fraction whose denominator is an integer. This process is still used today and is useful in other areas of mathematics, too.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Rationalizing the Denominator]

The process of converting a fraction with a radical in the denominator to an equivalent fraction whose denominator is an integer is called #strong[rationalizing the denominator].
]

Square roots of numbers that are not perfect squares are irrational numbers. When we #strong[rationalize the denominator], we write an equivalent fraction with a rational number in the denominator.

Let’s look at a numerical example.

#math.equation(block: true, alt: "Suppose we need an approximate value for the fraction., the fraction 1 over the square root of 2; A five decimal place approximation to the square root of 2 is 1.41421 ., the fraction 1 over 1.41421; Without a calculator, would you want to do this division?, 1.41421 1.0")[$"Suppose we need an approximate value for the fraction." &  &  & frac(1, sqrt(2)) \ "A five decimal place approximation to" #h(0.2em) sqrt(2) #h(0.2em) "is" #h(0.2em) 1.41421 . &  &  & frac(1, 1.41421) \ "Without a calculator, would you want to do this division?" &  &  & 1.41421 ") " overline(1.0)$]

But we can find a fraction equivalent to #math.equation(block: false, alt: "the fraction 1 over the square root of 2")[$frac(1, sqrt(2))$] by multiplying the numerator and denominator by #math.equation(block: false, alt: "the square root of 2")[$sqrt(2)$].

#figure(figph[This figure shows three fractions. The first fraction is 1 over the square root of 2. The second is 1 times the square root of 2 over the square root of 2 times the square root of 2. The third shows the square root of 2 over 2.], alt: "This figure shows three fractions. The first fraction is 1 over the square root of 2. The second is 1 times the square root of 2 over the square root of 2 times the square root of 2. The third shows the square root of 2 over 2.", caption: none)

Now if we need an approximate value, we divide #math.equation(block: false, alt: "2 1.41421")[$2 ") " overline(1.41421)$]. This is much easier.

Even though we have calculators available nearly everywhere, a fraction with a radical in the denominator still must be rationalized. It is not considered simplified if the denominator contains a square root.

Similarly, a square root is not considered simplified if the radicand contains a fraction.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Simplified Square Roots]

A square root is considered simplified if there are

- no perfect-square factors in the radicand
- no fractions in the radicand
- no square roots in the denominator of a fraction
]

To rationalize a denominator, we use the property that #math.equation(block: false, alt: "open parenthesis the square root of a close parenthesis squared equals a")[$attach(( sqrt(a) ), t: 2) = a$]. If we square an irrational square root, we get a rational number.

We will use this property to rationalize the denominator in the next example.

#examplebox("Example 7")[][
Simplify: #math.equation(block: false, alt: "the fraction 4 over the square root of 3")[$frac(4, sqrt(3))$].

#solutionbox[
To rationalize a denominator, we can multiply a square root by itself. To keep the fraction equivalent, we multiply both the numerator and denominator by the same factor.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 4 over the square root of 3")[$frac(4, sqrt(3))$]]),
  [Multiply both the numerator and denominator by #math.equation(block: false, alt: "the square root of 3 .")[$sqrt(3) .$]], [#math.equation(block: false, alt: "the fraction 4 times the square root of 3 over the square root of 3 times the square root of 3")[$frac(4 · sqrt(3), sqrt(3) · sqrt(3))$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 4 the square root of 3 over 3")[$frac(4 sqrt(3), 3)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 5 over the square root of 3")[$frac(5, sqrt(3))$].

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 the square root of 3 over 3")[$frac(5 sqrt(3), 3)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 6 over the square root of 5")[$frac(6, sqrt(5))$].

#solutionbox[
#math.equation(block: true, alt: "the fraction 6 the square root of 5 over 5")[$frac(6 sqrt(5), 5)$]
]
]

#examplebox("Example 8")[][
Simplify: #math.equation(block: false, alt: "minus the fraction 8 over 3 the square root of 6")[$− frac(8, 3 sqrt(6))$].

#solutionbox[
To remove the square root from the denominator, we multiply it by itself. To keep the fractions equivalent, we multiply both the numerator and denominator by #math.equation(block: false, alt: "the square root of 6")[$sqrt(6)$].

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression displaying a fraction with a negative sign, where the numerator is 8 and the denominator is 3 multiplied by the square root of 6, represented as -8 / (3sqrt(6)).], alt: "A mathematical expression displaying a fraction with a negative sign, where the numerator is 8 and the denominator is 3 multiplied by the square root of 6, represented as -8 / (3sqrt(6)).", caption: none)]),
  [Multiply both the numerator and the denominator by #math.equation(block: false, alt: "the square root of 6")[$sqrt(6)$].], [#figure(figph[A mathematical expression showing the process of rationalizing a denominator, where both the numerator and denominator are multiplied by the square root of 6.], alt: "A mathematical expression showing the process of rationalizing a denominator, where both the numerator and denominator are multiplied by the square root of 6.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical expression showing a negative fraction: - (8 times the square root of 6) divided by (3 multiplied by 6).], alt: "A mathematical expression showing a negative fraction: - (8 times the square root of 6) divided by (3 multiplied by 6).", caption: none)],
  [Remove common factors.], [#figure(figph[A mathematical expression showing a negative fraction where the factor 2 is crossed out in both the numerator and denominator. The numerator is 4 times the cancelled 2 times the square root of 6, and the denominator is 3 times the cancelled 2 times 3.], alt: "A mathematical expression showing a negative fraction where the factor 2 is crossed out in both the numerator and denominator. The numerator is 4 times the cancelled 2 times the square root of 6, and the denominator is 3 times the cancelled 2 times 3.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical expression showing negative four times the square root of six, all divided by nine. It's written as -4sqrt(6)/9.], alt: "A mathematical expression showing negative four times the square root of six, all divided by nine. It's written as -4sqrt(6)/9.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 5 over 2 the square root of 5")[$frac(5, 2 sqrt(5))$].

#solutionbox[
#math.equation(block: true, alt: "the fraction the square root of 5 over 2")[$frac(sqrt(5), 2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "minus the fraction 9 over 4 the square root of 3")[$− frac(9, 4 sqrt(3))$].

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 3 the square root of 3 over 4")[$− frac(3 sqrt(3), 4)$]
]
]

Always simplify the radical in the denominator first, before you rationalize it. This way the numbers stay smaller and easier to work with.

#examplebox("Example 9")[][
Simplify: #math.equation(block: false, alt: "the square root of the fraction 5 over 12")[$sqrt(frac(5, 12))$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression displays the square root of the fraction 5/12, with the numerator 5 and the denominator 12 clearly visible under the radical symbol.], alt: "A mathematical expression displays the square root of the fraction 5/12, with the numerator 5 and the denominator 12 clearly visible under the radical symbol.", caption: none)]),
  [The fraction is not a perfect square, so rewrite using the #linebreak() Quotient Property.], [#figure(figph[Mathematical expression: square root of 5 over square root of 12.], alt: "Mathematical expression: square root of 5 over square root of 12.", caption: none)],
  [Simplify the denominator], [#figure(figph[A mathematical fraction with the square root of 5 in the numerator and 2 times the square root of 3 in the denominator.], alt: "A mathematical fraction with the square root of 5 in the numerator and 2 times the square root of 3 in the denominator.", caption: none)],
  [Rationalize the denominator.], [#figure(figph[A mathematical expression illustrating the process of rationalizing a denominator, specifically multiplying both the numerator and denominator by the square root of 3.], alt: "A mathematical expression illustrating the process of rationalizing a denominator, specifically multiplying both the numerator and denominator by the square root of 3.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical fraction with the square root of 15 in the numerator and 2 multiplied by 3 in the denominator.], alt: "A mathematical fraction with the square root of 15 in the numerator and 2 multiplied by 3 in the denominator.", caption: none)],
  [Simplify.], [#figure(figph[The mathematical expression showing the square root of 15, all divided by 6.], alt: "The mathematical expression showing the square root of 15, all divided by 6.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of the fraction 7 over 18")[$sqrt(frac(7, 18))$].

#solutionbox[
#math.equation(block: true, alt: "the fraction the square root of 14 over 6")[$frac(sqrt(14), 6)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of the fraction 3 over 32")[$sqrt(frac(3, 32))$].

#solutionbox[
#math.equation(block: true, alt: "the fraction the square root of 6 over 8")[$frac(sqrt(6), 8)$]
]
]

#examplebox("Example 10")[][
Simplify: #math.equation(block: false, alt: "the square root of the fraction 11 over 28")[$sqrt(frac(11, 28))$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression showing the square root of the fraction 11 over 28.], alt: "A mathematical expression showing the square root of the fraction 11 over 28.", caption: none)]),
  [Rewrite using the Quotient Property.], [#figure(figph[A mathematical expression showing the square root of 11 divided by the square root of 28, presented as a fraction.], alt: "A mathematical expression showing the square root of 11 divided by the square root of 28, presented as a fraction.", caption: none)],
  [Simplify the denominator.], [#figure(figph[A mathematical expression showing the square root of 11 divided by 2 times the square root of 7, represented as sqrt(11) / (2\*sqrt(7)).], alt: "A mathematical expression showing the square root of 11 divided by 2 times the square root of 7, represented as sqrt(11) / (2*sqrt(7)).", caption: none)],
  [Rationalize the denominator.], [#figure(figph[Step in simplifying a radical expression by multiplying numerator and denominator by the square root of 7.], alt: "Step in simplifying a radical expression by multiplying numerator and denominator by the square root of 7.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical expression displaying the square root of 77 divided by the product of 2 and 7.], alt: "A mathematical expression displaying the square root of 77 divided by the product of 2 and 7.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical expression showing the square root of 77 divided by 14. The numerator is the square root of 77, and the denominator is 14.], alt: "A mathematical expression showing the square root of 77 divided by 14. The numerator is the square root of 77, and the denominator is 14.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of the fraction 3 over 27")[$sqrt(frac(3, 27))$].

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 3")[$frac(1, 3)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of the fraction 10 over 50")[$sqrt(frac(10, 50))$].

#solutionbox[
#math.equation(block: true, alt: "the fraction the square root of 5 over 5")[$frac(sqrt(5), 5)$]
]
]


=== Rationalize a Two-Term Denominator

When the denominator of a fraction is a sum or difference with square roots, we use the Product of Conjugates pattern to rationalize the denominator.

#math.equation(block: true, alt: "open parenthesis a minus b close parenthesis open parenthesis a plus b close parenthesis, open parenthesis 2 minus the square root of 5 close parenthesis open parenthesis 2 plus the square root of 5 close parenthesis; a squared minus b squared, 2 squared minus open parenthesis the square root of 5 close parenthesis squared; 4 minus 5; −1")[$( a − b ) ( a + b ) &  &  & #h(2em) ( 2 − sqrt(5) ) ( 2 + sqrt(5) ) \ a^(2) − b^(2) &  &  & #h(2em) 2^(2) − attach(( sqrt(5) ), t: 2) \  &  &  & #h(2em) 4 − 5 \  &  &  & #h(2em) −1$]When we multiply a binomial that includes a square root by its conjugate, the product has no square roots.

#examplebox("Example 11")[][
Simplify: #math.equation(block: false, alt: "the fraction 4 over 4 plus the square root of 2")[$frac(4, 4 + sqrt(2))$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A fraction with 4 in the numerator and 4 + square root of 2 in the denominator.], alt: "A fraction with 4 in the numerator and 4 + square root of 2 in the denominator.", caption: none)]),
  [Multiply the numerator and denominator by the conjugate of the denominator.], [#figure(figph[A mathematical expression showing the fraction 4(4-sqrt(2)) divided by the product of (4+sqrt(2)) and (4-sqrt(2)), with (4-sqrt(2)) highlighted in red in both the numerator and denominator.], alt: "A mathematical expression showing the fraction 4(4-sqrt(2)) divided by the product of (4+sqrt(2)) and (4-sqrt(2)), with (4-sqrt(2)) highlighted in red in both the numerator and denominator.", caption: none)],
  [Multiply the conjugates in the denominator.], [#figure(figph[A mathematical fraction is shown. The numerator is 4(4 - square root of 2). The denominator is 4 squared - (square root of 2) squared.], alt: "A mathematical fraction is shown. The numerator is 4(4 - square root of 2). The denominator is 4 squared - (square root of 2) squared.", caption: none)],
  [Simplify the denominator.], [#figure(figph[A mathematical expression showing a fraction. The numerator is 4(4 - square root of 2), and the denominator is 16 - 2.], alt: "A mathematical expression showing a fraction. The numerator is 4(4 - square root of 2), and the denominator is 16 - 2.", caption: none)],
  [Simplify the denominator.], [#figure(figph[A mathematical expression showing the fraction 4 multiplied by (4 minus the square root of 2), all divided by 14.], alt: "A mathematical expression showing the fraction 4 multiplied by (4 minus the square root of 2), all divided by 14.", caption: none)],
  [Remove common factors from the numerator and denominator.], [#figure(figph[A mathematical expression representing the fraction 2 multiplied by the quantity 4 minus the square root of 2, all divided by 7.], alt: "A mathematical expression representing the fraction 2 multiplied by the quantity 4 minus the square root of 2, all divided by 7.", caption: none)],
  [We leave the numerator in factored form to make it easier to look for common factors after we have simplified the denominator.], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 2 over 2 plus the square root of 3")[$frac(2, 2 + sqrt(3))$].

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 open parenthesis 2 minus the square root of 3 close parenthesis over 1")[$frac(2 ( 2 − sqrt(3) ), 1)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 5 over 5 plus the square root of 3")[$frac(5, 5 + sqrt(3))$].

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 open parenthesis 5 minus the square root of 3 close parenthesis over 22")[$frac(5 ( 5 − sqrt(3) ), 22)$]
]
]

#examplebox("Example 12")[][
Simplify: #math.equation(block: false, alt: "the fraction 5 over 2 minus the square root of 3")[$frac(5, 2 − sqrt(3))$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical fraction is displayed, with 5 as the numerator and (2 - sqrt(3)) as the denominator. This represents the expression 5 / (2 minus the square root of 3).], alt: "A mathematical fraction is displayed, with 5 as the numerator and (2 - sqrt(3)) as the denominator. This represents the expression 5 / (2 minus the square root of 3).", caption: none)]),
  [Multiply the numerator and denominator by the conjugate of the denominator.], [#figure(figph[A fraction: 5(2 + sqrt(3)) / ((2 - sqrt(3))(2 + sqrt(3))). The common term (2 + sqrt(3)) is highlighted in red in both the numerator and denominator, indicating a step in simplification.], alt: "A fraction: 5(2 + sqrt(3)) / ((2 - sqrt(3))(2 + sqrt(3))). The common term (2 + sqrt(3)) is highlighted in red in both the numerator and denominator, indicating a step in simplification.", caption: none)],
  [Multiply the conjugates in the denominator.], [#figure(figph[A mathematical expression showing a fraction. The numerator is 5 times (2 plus the square root of 3). The denominator is 2 squared minus the square root of 3 squared.], alt: "A mathematical expression showing a fraction. The numerator is 5 times (2 plus the square root of 3). The denominator is 2 squared minus the square root of 3 squared.", caption: none)],
  [Simplify the denominator.], [#figure(figph[A mathematical expression showing a fraction with 5 multiplied by (2 plus the square root of 3) in the numerator, and (4 minus 3) in the denominator, on a white background.], alt: "A mathematical expression showing a fraction with 5 multiplied by (2 plus the square root of 3) in the numerator, and (4 minus 3) in the denominator, on a white background.", caption: none)],
  [Simplify the denominator.], [#figure(figph[A mathematical expression showing the fraction 5(2 +  3)/1.], alt: "A mathematical expression showing the fraction 5(2 + 3)/1.", caption: none)],
  [Simplify.], [#figure(figph[The mathematical expression 5 multiplied by the sum of 2 and the square root of 3 is shown on a white background. It represents 5(2 + 'sqrt'(3)).], alt: "The mathematical expression 5 multiplied by the sum of 2 and the square root of 3 is shown on a white background. It represents 5(2 + 'sqrt'(3)).", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 3 over 1 minus the square root of 5")[$frac(3, 1 − sqrt(5))$].

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 3 open parenthesis 1 plus the square root of 5 close parenthesis over 4")[$− frac(3 ( 1 + sqrt(5) ), 4)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 2 over 4 minus the square root of 6")[$frac(2, 4 − sqrt(6))$].

#solutionbox[
#math.equation(block: true, alt: "the fraction 4 plus the square root of 6 over 5")[$frac(4 + sqrt(6), 5)$]
]
]

#examplebox("Example 13")[][
Simplify: #math.equation(block: false, alt: "the fraction the square root of 3 over the square root of u minus the square root of 6")[$frac(sqrt(3), sqrt(u) − sqrt(6))$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression showing a fraction with the square root of 3 in the numerator and the square root of u minus the square root of 6 in the denominator.], alt: "A mathematical expression showing a fraction with the square root of 3 in the numerator and the square root of u minus the square root of 6 in the denominator.", caption: none)]),
  [Multiply the numerator and denominator by the conjugate of the denominator.], [#figure(figph[A fraction with a square root of three times the quantity of the square root of u plus the square root of six in the numerator. The denominator is the product of the quantity of the square root of u minus the square root of six and the quantity of the square root of u plus the square root of six.], alt: "A fraction with a square root of three times the quantity of the square root of u plus the square root of six in the numerator. The denominator is the product of the quantity of the square root of u minus the square root of six and the quantity of the square root of u plus the square root of six.", caption: none)],
  [Multiply the conjugates in the denominator.], [#figure(figph[A mathematical expression showing the fraction: square root of 3 multiplied by the sum of square root of 'u' and square root of 6, all divided by 'u' minus 6.], alt: "A mathematical expression showing the fraction: square root of 3 multiplied by the sum of square root of 'u' and square root of 6, all divided by 'u' minus 6.", caption: none)],
  [Simplify the denominator.], [#figure(figph[A fraction with a numerator of 'square root 3 times the quantity square root u plus square root 6' and a denominator of 'u minus 6'.], alt: "A fraction with a numerator of 'square root 3 times the quantity square root u plus square root 6' and a denominator of 'u minus 6'.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction the square root of 5 over the square root of x plus the square root of 2")[$frac(sqrt(5), sqrt(x) + sqrt(2))$].

#solutionbox[
#math.equation(block: true, alt: "the fraction the square root of 5 open parenthesis the square root of x minus the square root of 2 close parenthesis over x minus 2")[$frac(sqrt(5) ( sqrt(x) − sqrt(2) ), x − 2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction the square root of 10 over the square root of y minus the square root of 3")[$frac(sqrt(10), sqrt(y) − sqrt(3))$].

#solutionbox[
#math.equation(block: true, alt: "the fraction the square root of 10 open parenthesis the square root of y plus the square root of 3 close parenthesis over y minus 3")[$frac(sqrt(10) ( sqrt(y) + sqrt(3) ), y − 3)$]
]
]

#examplebox("Example 14")[][
Simplify: #math.equation(block: false, alt: "the fraction the square root of x plus the square root of 7 over the square root of x minus the square root of 7")[$frac(sqrt(x) + sqrt(7), sqrt(x) − sqrt(7))$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical fraction is displayed, with the numerator as the square root of x plus the square root of 7, and the denominator as the square root of x minus the square root of 7.], alt: "A mathematical fraction is displayed, with the numerator as the square root of x plus the square root of 7, and the denominator as the square root of x minus the square root of 7.", caption: none)]),
  [Multiply the numerator and denominator by the conjugate of the denominator.], [#figure(figph[An algebraic fraction showing (sqrt(x)+sqrt(7))^2 divided by (sqrt(x)-sqrt(7))(sqrt(x)+sqrt(7)), illustrating a step in simplifying radical expressions.], alt: "An algebraic fraction showing (sqrt(x)+sqrt(7))^2 divided by (sqrt(x)-sqrt(7))(sqrt(x)+sqrt(7)), illustrating a step in simplifying radical expressions.", caption: none)],
  [Multiply the conjugates in the denominator.], [#figure(figph[A mathematical fraction with the numerator as the product of (square root x + square root 7) and (square root x + square root 7), and the denominator as (square root x) squared minus (square root 7) squared.], alt: "A mathematical fraction with the numerator as the product of (square root x + square root 7) and (square root x + square root 7), and the denominator as (square root x) squared minus (square root 7) squared.", caption: none)],
  [Simplify the denominator.], [#figure(figph[A fraction with numerator (sqrt(x) + sqrt(7))^2 and denominator (x - 7).], alt: "A fraction with numerator (sqrt(x) + sqrt(7))^2 and denominator (x - 7).", caption: none)],
  [We do not square the numerator. In factored form, we can see there are no common factors to remove from the numerator and denominator.], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction the square root of p plus the square root of 2 over the square root of p minus the square root of 2")[$frac(sqrt(p) + sqrt(2), sqrt(p) − sqrt(2))$].

#solutionbox[
#math.equation(block: true, alt: "the fraction open parenthesis the square root of p plus the square root of 2 close parenthesis over p minus 2 squared")[$attach(frac(( sqrt(p) + sqrt(2) ), p − 2), t: 2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction the square root of q minus the square root of 10 over the square root of q plus the square root of 10")[$frac(sqrt(q) − sqrt(10), sqrt(q) + sqrt(10))$].

#solutionbox[
#math.equation(block: true, alt: "the fraction open parenthesis the square root of q minus the square root of 10 close parenthesis over q minus 10 squared")[$attach(frac(( sqrt(q) − sqrt(10) ), q − 10), t: 2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Access this online resource for additional instruction and practice with dividing and rationalizing.

- #link("https://openstax.org/l/25DivideRation")[Dividing and Rationalizing]
]


=== Key Concepts

- #strong[Quotient Property of Square Roots]
- If #emph[a], #emph[b] are non-negative real numbers and #math.equation(block: false, alt: "b not equal to 0")[$b ≠ 0$], then
#math.equation(block: true, alt: "the square root of the fraction a over b equals the fraction the square root of a over the square root of b and the fraction the square root of a over the square root of b equals the square root of the fraction a over b")[$sqrt(frac(a, b)) = frac(sqrt(a), sqrt(b)) #h(1em) "and" #h(1em) frac(sqrt(a), sqrt(b)) = sqrt(frac(a, b))$]
- #strong[Simplified Square Roots] #linebreak() A square root is considered simplified if there are
- no perfect square factors in the radicand
- no fractions in the radicand
- no square roots in the denominator of a fraction


==== Practice Makes Perfect

#strong[Divide Square Roots]

In the following exercises, simplify.

#math.equation(block: true, alt: "the fraction the square root of 27 over 6")[$frac(sqrt(27), 6)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction the square root of 3 over 2")[$frac(sqrt(3), 2)$]
]

#math.equation(block: true, alt: "the fraction the square root of 50 over 10")[$frac(sqrt(50), 10)$]

#math.equation(block: true, alt: "the fraction the square root of 72 over 9")[$frac(sqrt(72), 9)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 the square root of 2 over 3")[$frac(2 sqrt(2), 3)$]
]

#math.equation(block: true, alt: "the fraction the square root of 243 over 6")[$frac(sqrt(243), 6)$]

#math.equation(block: true, alt: "the fraction 2 minus the square root of 32 over 8")[$frac(2 − sqrt(32), 8)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 minus 2 the square root of 2 over 4")[$frac(1 − 2 sqrt(2), 4)$]
]

#math.equation(block: true, alt: "the fraction 3 plus the square root of 27 over 9")[$frac(3 + sqrt(27), 9)$]

#math.equation(block: true, alt: "the fraction 6 plus the square root of 45 over 6")[$frac(6 + sqrt(45), 6)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 plus the square root of 5 over 2")[$frac(2 + sqrt(5), 2)$]
]

#math.equation(block: true, alt: "the fraction 10 minus the square root of 200 over 20")[$frac(10 − sqrt(200), 20)$]

#math.equation(block: true, alt: "the fraction the square root of 80 over the square root of 125")[$frac(sqrt(80), sqrt(125))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 4 over 5")[$frac(4, 5)$]
]

#math.equation(block: true, alt: "the fraction the square root of 72 over the square root of 200")[$frac(sqrt(72), sqrt(200))$]

#math.equation(block: true, alt: "the fraction the square root of 128 over the square root of 72")[$frac(sqrt(128), sqrt(72))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 4 over 3")[$frac(4, 3)$]
]

#math.equation(block: true, alt: "the fraction the square root of 48 over the square root of 75")[$frac(sqrt(48), sqrt(75))$]

ⓐ #math.equation(block: false, alt: "the fraction the square root of 8 x to the power 6 over the square root of 2 x squared")[$frac(sqrt(8 x^(6)), sqrt(2 x^(2)))$] ⓑ #math.equation(block: false, alt: "the fraction the square root of 200 m to the power 5 over the square root of 98 m")[$frac(sqrt(200 m^(5)), sqrt(98 m))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "2 x squared")[$2 x^(2)$] ⓑ #math.equation(block: false, alt: "the fraction 10 m squared over 7")[$frac(10 m^(2), 7)$]
]

ⓐ #math.equation(block: false, alt: "the fraction the square root of 10 y cubed over the square root of 5 y")[$frac(sqrt(10 y^(3)), sqrt(5 y))$] ⓑ #math.equation(block: false, alt: "the fraction the square root of 108 n to the power 7 over the square root of 243 n cubed")[$frac(sqrt(108 n^(7)), sqrt(243 n^(3)))$]

#math.equation(block: true, alt: "the fraction the square root of 75 r cubed over the square root of 108 r")[$frac(sqrt(75 r^(3)), sqrt(108 r))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 r over 6")[$frac(5 r, 6)$]
]

#math.equation(block: true, alt: "the fraction the square root of 196 q to the power 5 over the square root of 484 q")[$frac(sqrt(196 q^(5)), sqrt(484 q))$]

#math.equation(block: true, alt: "the fraction the square root of 108 p to the power 5 q squared over the square root of 3 p cubed q to the power 6")[$frac(sqrt(108 p^(5) q^(2)), sqrt(3 p^(3) q^(6)))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 6 p the square root of 102 over q squared")[$frac(6 p sqrt(102), q^(2))$]
]

#math.equation(block: true, alt: "the fraction the square root of 98 r s to the power 10 over the square root of 2 r cubed s to the power 4")[$frac(sqrt(98 r s^(10)), sqrt(2 r^(3) s^(4)))$]

#math.equation(block: true, alt: "the fraction the square root of 320 m n to the power 5 over the square root of 45 m to the power 7 n cubed")[$frac(sqrt(320 m n^(5)), sqrt(45 m^(7) n^(3)))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 8 n over 3 m cubed")[$frac(8 n, 3 m^(3))$]
]

#math.equation(block: true, alt: "the fraction the square root of 810 c cubed d to the power 7 over the square root of 1000 c to the power 5 d")[$frac(sqrt(810 c^(3) d^(7)), sqrt(1000 c^(5) d))$]

#math.equation(block: true, alt: "the fraction the square root of 98 over 14")[$frac(sqrt(98), 14)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction the square root of 2 over 2")[$frac(sqrt(2), 2)$]
]

#math.equation(block: true, alt: "the fraction the square root of 72 over 18")[$frac(sqrt(72), 18)$]

#math.equation(block: true, alt: "the fraction 5 plus the square root of 125 over 15")[$frac(5 + sqrt(125), 15)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 plus the square root of 5 over 3")[$frac(1 + sqrt(5), 3)$]
]

#math.equation(block: true, alt: "the fraction 6 minus the square root of 45 over 12")[$frac(6 − sqrt(45), 12)$]

#math.equation(block: true, alt: "the fraction the square root of 96 over the square root of 150")[$frac(sqrt(96), sqrt(150))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 4 over 5")[$frac(4, 5)$]
]

#math.equation(block: true, alt: "the fraction the square root of 28 over the square root of 63")[$frac(sqrt(28), sqrt(63))$]

#math.equation(block: true, alt: "the fraction the square root of 26 y to the power 7 over the square root of 2 y")[$frac(sqrt(26 y^(7)), sqrt(2 y))$]

#solutionbox[
#math.equation(block: true, alt: "y cubed the square root of 13")[$y^(3) sqrt(13)$]
]

#math.equation(block: true, alt: "the fraction the square root of 15 x cubed over the square root of 3 x")[$frac(sqrt(15 x^(3)), sqrt(3 x))$]

#strong[Rationalize a One-Term Denominator]

In the following exercises, simplify and rationalize the denominator.

#math.equation(block: true, alt: "the fraction 10 over the square root of 6")[$frac(10, sqrt(6))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 the square root of 6 over 3")[$frac(5 sqrt(6), 3)$]
]

#math.equation(block: true, alt: "the fraction 8 over the square root of 3")[$frac(8, sqrt(3))$]

#math.equation(block: true, alt: "the fraction 6 over the square root of 7")[$frac(6, sqrt(7))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 6 the square root of 7 over 7")[$frac(6 sqrt(7), 7)$]
]

#math.equation(block: true, alt: "the fraction 4 over the square root of 5")[$frac(4, sqrt(5))$]

#math.equation(block: true, alt: "the fraction 3 over the square root of 13")[$frac(3, sqrt(13))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 the square root of 13 over 13")[$frac(3 sqrt(13), 13)$]
]

#math.equation(block: true, alt: "the fraction 10 over the square root of 11")[$frac(10, sqrt(11))$]

#math.equation(block: true, alt: "the fraction 10 over 3 the square root of 10")[$frac(10, 3 sqrt(10))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction the square root of 10 over 3")[$frac(sqrt(10), 3)$]
]

#math.equation(block: true, alt: "the fraction 2 over 5 the square root of 2")[$frac(2, 5 sqrt(2))$]

#math.equation(block: true, alt: "the fraction 4 over 9 the square root of 5")[$frac(4, 9 sqrt(5))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 4 the square root of 5 over 45")[$frac(4 sqrt(5), 45)$]
]

#math.equation(block: true, alt: "the fraction 9 over 2 the square root of 7")[$frac(9, 2 sqrt(7))$]

#math.equation(block: true, alt: "minus the fraction 9 over 2 the square root of 3")[$− frac(9, 2 sqrt(3))$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 3 the square root of 3 over 2")[$− frac(3 sqrt(3), 2)$]
]

#math.equation(block: true, alt: "minus the fraction 8 over 3 the square root of 6")[$− frac(8, 3 sqrt(6))$]

#math.equation(block: true, alt: "the square root of the fraction 3 over 20")[$sqrt(frac(3, 20))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction the square root of 15 over 10")[$frac(sqrt(15), 10)$]
]

#math.equation(block: true, alt: "the square root of the fraction 4 over 27")[$sqrt(frac(4, 27))$]

#math.equation(block: true, alt: "the square root of the fraction 7 over 40")[$sqrt(frac(7, 40))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction the square root of 70 over 20")[$frac(sqrt(70), 20)$]
]

#math.equation(block: true, alt: "the square root of the fraction 8 over 45")[$sqrt(frac(8, 45))$]

#math.equation(block: true, alt: "the square root of the fraction 19 over 175")[$sqrt(frac(19, 175))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction the square root of 133 over 35")[$frac(sqrt(133), 35)$]
]

#math.equation(block: true, alt: "the square root of the fraction 17 over 192")[$sqrt(frac(17, 192))$]

#strong[Rationalize a Two-Term Denominator]

In the following exercises, simplify by rationalizing the denominator.

ⓐ #math.equation(block: false, alt: "the fraction 3 over 3 plus the square root of 11")[$frac(3, 3 + sqrt(11))$] ⓑ #math.equation(block: false, alt: "the fraction 8 over 1 minus the square root of 5")[$frac(8, 1 − sqrt(5))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "the fraction 3 open parenthesis 3 minus the square root of 11 close parenthesis over −2")[$frac(3 ( 3 − sqrt(11) ), −2)$] ⓑ #math.equation(block: false, alt: "−2 open parenthesis 1 plus the square root of 5 close parenthesis")[$−2 ( 1 + sqrt(5) )$]
]

ⓐ #math.equation(block: false, alt: "the fraction 4 over 4 plus the square root of 7")[$frac(4, 4 + sqrt(7))$] ⓑ #math.equation(block: false, alt: "the fraction 7 over 2 minus the square root of 6")[$frac(7, 2 − sqrt(6))$]

ⓐ #math.equation(block: false, alt: "the fraction 5 over 5 plus the square root of 6")[$frac(5, 5 + sqrt(6))$] ⓑ #math.equation(block: false, alt: "the fraction 6 over 3 minus the square root of 7")[$frac(6, 3 − sqrt(7))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "the fraction 5 open parenthesis 5 minus the square root of 6 close parenthesis over 19")[$frac(5 ( 5 − sqrt(6) ), 19)$] ⓑ #math.equation(block: false, alt: "3 open parenthesis 3 plus the square root of 7 close parenthesis")[$3 ( 3 + sqrt(7) )$]
]

ⓐ #math.equation(block: false, alt: "the fraction 6 over 6 plus the square root of 5")[$frac(6, 6 + sqrt(5))$] ⓑ #math.equation(block: false, alt: "the fraction 5 over 4 minus the square root of 11")[$frac(5, 4 − sqrt(11))$]

#math.equation(block: true, alt: "the fraction the square root of 3 over the square root of m minus the square root of 5")[$frac(sqrt(3), sqrt(m) − sqrt(5))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction the square root of 3 open parenthesis the square root of m plus the square root of 5 close parenthesis over m minus 5")[$frac(sqrt(3) ( sqrt(m) + sqrt(5) ), m − 5)$]
]

#math.equation(block: true, alt: "the fraction the square root of 5 over the square root of n minus the square root of 7")[$frac(sqrt(5), sqrt(n) − sqrt(7))$]

#math.equation(block: true, alt: "the fraction the square root of 2 over the square root of x minus the square root of 6")[$frac(sqrt(2), sqrt(x) − sqrt(6))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction the square root of 2 open parenthesis the square root of x plus the square root of 6 close parenthesis over x minus 6")[$frac(sqrt(2) ( sqrt(x) + sqrt(6) ), x − 6)$]
]

#math.equation(block: true, alt: "the fraction the square root of 7 over the square root of y plus the square root of 3")[$frac(sqrt(7), sqrt(y) + sqrt(3))$]

#math.equation(block: true, alt: "the fraction the square root of r plus the square root of 5 over the square root of r minus the square root of 5")[$frac(sqrt(r) + sqrt(5), sqrt(r) − sqrt(5))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction open parenthesis the square root of r plus the square root of 5 close parenthesis over r minus 5 squared")[$attach(frac(( sqrt(r) + sqrt(5) ), r − 5), t: 2)$]
]

#math.equation(block: true, alt: "the fraction the square root of s minus the square root of 6 over the square root of s plus the square root of 6")[$frac(sqrt(s) − sqrt(6), sqrt(s) + sqrt(6))$]

#math.equation(block: true, alt: "the fraction the square root of 150 x squared y to the power 6 over the square root of 6 x to the power 4 y squared")[$frac(sqrt(150 x^(2) y^(6)), sqrt(6 x^(4) y^(2)))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 y squared over x")[$frac(5 y^(2), x)$]
]

#math.equation(block: true, alt: "the fraction the square root of 80 p cubed q over the square root of 5 p q to the power 5")[$frac(sqrt(80 p^(3) q), sqrt(5 p q^(5)))$]

#math.equation(block: true, alt: "the fraction 15 over the square root of 5")[$frac(15, sqrt(5))$]

#solutionbox[
#math.equation(block: true, alt: "3 the square root of 5")[$3 sqrt(5)$]
]

#math.equation(block: true, alt: "the fraction 3 over 5 the square root of 8")[$frac(3, 5 sqrt(8))$]

#math.equation(block: true, alt: "the square root of the fraction 8 over 54")[$sqrt(frac(8, 54))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 the square root of 3 over 9")[$frac(2 sqrt(3), 9)$]
]

#math.equation(block: true, alt: "the square root of the fraction 12 over 20")[$sqrt(frac(12, 20))$]

#math.equation(block: true, alt: "the fraction 3 over 5 plus the square root of 5")[$frac(3, 5 + sqrt(5))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 open parenthesis 5 minus the square root of 5 close parenthesis over 20")[$frac(3 ( 5 − sqrt(5) ), 20)$]
]

#math.equation(block: true, alt: "the fraction 20 over 4 minus the square root of 3")[$frac(20, 4 − sqrt(3))$]

#math.equation(block: true, alt: "the fraction the square root of 2 over the square root of x minus the square root of 3")[$frac(sqrt(2), sqrt(x) − sqrt(3))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction the square root of 2 open parenthesis the square root of x plus the square root of 3 close parenthesis over x minus 3")[$frac(sqrt(2) ( sqrt(x) + sqrt(3) ), x − 3)$]
]

#math.equation(block: true, alt: "the fraction the square root of 5 over the square root of y minus the square root of 7")[$frac(sqrt(5), sqrt(y) − sqrt(7))$]

#math.equation(block: true, alt: "the fraction the square root of x plus the square root of 8 over the square root of x minus the square root of 8")[$frac(sqrt(x) + sqrt(8), sqrt(x) − sqrt(8))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction open parenthesis the square root of x plus 2 the square root of 2 close parenthesis over x minus 8 squared")[$attach(frac(( sqrt(x) + 2 sqrt(2) ), x − 8), t: 2)$]
]

#math.equation(block: true, alt: "the fraction the square root of m minus the square root of 3 over the square root of m plus the square root of 3")[$frac(sqrt(m) − sqrt(3), sqrt(m) + sqrt(3))$]


==== Everyday Math

A supply kit is dropped from an airplane flying at an altitude of 250 feet. Simplify #math.equation(block: false, alt: "the square root of the fraction 250 over 16")[$sqrt(frac(250, 16))$] to determine how many seconds it takes for the supply kit to reach the ground.

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 the square root of 10 over 4 seconds")[$frac(5 sqrt(10), 4) #h(0.2em) "seconds"$]
]

A flare is dropped into the ocean from an airplane flying at an altitude of 1,200 feet. Simplify #math.equation(block: false, alt: "the square root of the fraction 1200 over 16")[$sqrt(frac(1200, 16))$] to determine how many seconds it takes for the flare to reach the ocean.


==== Writing Exercises

+ ⓐ Simplify #math.equation(block: false, alt: "the square root of the fraction 27 over 3")[$sqrt(frac(27, 3))$] and explain all your steps.
+ ⓑ Simplify #math.equation(block: false, alt: "the square root of the fraction 27 over 5")[$sqrt(frac(27, 5))$] and explain all your steps.
+ ⓒ Why are the two methods of simplifying square roots different?

#solutionbox[
Answers will vary.
]

+ ⓐ Approximate #math.equation(block: false, alt: "the fraction 1 over the square root of 2")[$frac(1, sqrt(2))$] by dividing #math.equation(block: false, alt: "the fraction 1 over 1.414")[$frac(1, 1.414)$] using long division without a calculator.
+ ⓑ Rationalizing the denominator of #math.equation(block: false, alt: "the fraction 1 over the square root of 2")[$frac(1, sqrt(2))$] gives #math.equation(block: false, alt: "the fraction the square root of 2 over 2 .")[$frac(sqrt(2), 2) .$] Approximate #math.equation(block: false, alt: "the fraction the square root of 2 over 2")[$frac(sqrt(2), 2)$] by dividing #math.equation(block: false, alt: "the fraction 1.414 over 2")[$frac(1.414, 2)$] using long division without a calculator.
+ ⓒ Do you agree that rationalizing the denominator makes calculations easier? Why or why not?


==== Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

#figure(figph[This table has four columns and four rows. The columns are labeled, “I can…,” “confidently.,” “with some help.,” and “no – I don’t get it!” The rows under the column “I can…” read, “divide square roots,” “rationalize a one term denominator.,” and “rationalize a two term denominator.” All the other rows under the columns are empty.], alt: "This table has four columns and four rows. The columns are labeled, “I can…,” “confidently.,” “with some help.,” and “no – I don’t get it!” The rows under the column “I can…” read, “divide square roots,” “rationalize a one term denominator.,” and “rationalize a two term denominator.” All the other rows under the columns are empty.", caption: none)

ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?