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== 10.1#h(0.6em)Numbers and Operations


    
      
=== Order of Operations


      Numerical calculations often involve more than one operation. So that everyone agrees on how such expressions should be evaluated, we follow the #strong[order of operations].


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Order of Operations]


        + Simplify any expressions within grouping symbols (parentheses, brackets, square root bars, or fraction bars). Start with the innermost grouping symbols and work outward.
+ Evaluate all powers and roots.
+ Perform multiplications and divisions in order from left to right.
+ Perform additions and subtractions in order from left to right.
]


    
    
      
=== Parentheses and Fraction Bars


      We can use parentheses to override the multiplication-first rule. Compare the two expressions below.


      #math.equation(block: true, alt: "The sum of 4 times 6 and 10, 4 times 6 plus 10; 4 times the sum of 6 and 10, 4 open parenthesis 6 plus 10 close parenthesis")[$& "The sum of 4 times 6 and 10" &  & 4 ⋅ 6 + 10 \  & "4 times the sum of 6 and 10" &  & 4 ( 6 + 10 )$]
      In the first expression, we perform the multiplication #math.equation(block: false, alt: "4 times 6")[$4 × 6$] first, but in the second expression we perform the addition #math.equation(block: false, alt: "6 plus 10")[$6 + 10$] first, because it is enclosed in parentheses.


      The location (or absence) of parentheses can drastically alter the meaning of an expression. In the following example, note how the location of the parentheses changes the value of the expression.


      #examplebox("Example 1")[][
+ #math.equation(block: false, alt: "5 minus 3 times 4 squared, equals 5 minus 3 times 16; equals 5 minus 48 equals minus 43")[$5 − 3 ⋅ 4^(2) & = 5 − 3 ⋅ 16 \  & = 5 − 48 = − 43$]
+ #math.equation(block: false, alt: "5 minus open parenthesis 3 times 4 close parenthesis squared, equals 5 minus 12 squared; equals 5 minus 144 equals minus 139")[$5 − ( 3 ⋅ 4 )^(2) & = 5 − 12^(2) \  & = 5 − 144 = − 139$]
+ #math.equation(block: false, alt: "open parenthesis 5 minus 3 times 4 close parenthesis squared, equals open parenthesis 5 minus 12 close parenthesis squared; equals open parenthesis minus 7 close parenthesis squared equals 49")[$( 5 − 3 ⋅ 4 )^(2) & = ( 5 − 12 )^(2) \  & = ( − 7 )^(2) = 49$]
+ #math.equation(block: false, alt: "open parenthesis 5 minus 3 close parenthesis times 4 squared, equals 2 times 4 squared; equals 2 times 16 equals 32")[$( 5 − 3 ) ⋅ 4^(2) & = 2 ⋅ 4^(2) \  & = 2 ⋅ 16 = 32$]
]


      #notebox("Caution", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
In the expression #math.equation(block: false, alt: "5 minus 12 squared")[$5 − 12^(2)$], which appears in Example, the exponent #math.equation(block: false, alt: "2")[$2$] applies only to #math.equation(block: false, alt: "12")[$12$], not to #math.equation(block: false, alt: "minus 12")[$− 12$]. Thus, #math.equation(block: false, alt: "5 minus 12 squared not equal to 5 plus 144")[$5 − 12^(2) ≠ 5 + 144$].
]


      The order of operations mentions other grouping devices besides parentheses: fraction bars and square root bars. Notice how the placement of the fraction bar affects the expressions in the next example.


      #examplebox("Example 2")[][
+ #math.equation(block: false, alt: "the fraction 1 plus 2 over 3 times 4, equals the fraction 3 over 12; equals the fraction 1 over 4")[$frac(1 + 2, 3 ⋅ 4) & = frac(3, 12) \  & = frac(1, 4)$]
+ #math.equation(block: false, alt: "1 plus the fraction 2 over 3 times 4, equals 1 plus the fraction 2 over 12; equals 1 plus the fraction 1 over 6 equals the fraction 7 over 6")[$1 + frac(2, 3 ⋅ 4) & = 1 + frac(2, 12) \  & = 1 + frac(1, 6) = frac(7, 6)$]
+ #math.equation(block: false, alt: "the fraction 1 plus 2 over 3 times 4, equals the fraction 3 over 3 times 4; equals 1 times 4 equals 4")[$frac(1 + 2, 3) ⋅ 4 & = frac(3, 3) ⋅ 4 \  & = 1 ⋅ 4 = 4$]
+ #math.equation(block: false, alt: "1 plus the fraction 2 over 3 times 4, equals 1 plus the fraction 8 over 3; equals the fraction 3 over 3 plus the fraction 8 over 3 equals the fraction 11 over 3")[$1 + frac(2, 3) ⋅ 4 & = 1 + frac(8, 3) \  & = frac(3, 3) + frac(8, 3) = frac(11, 3)$]
]


    
    
      
=== Radicals


      You are already familiar with square roots. Every nonnegative number has two square roots, defined as follows:


      #math.equation(block: true, alt: "s is a square root of n if s squared equals n")[$bold(italic(s)) " " " " " is a square root of " " " " " bold(italic(n)) " " " " " if " " " " " bold(italic(s))^(2) = bold(italic(n))$]
      There are several other kinds of roots, one of which is called the #strong[cube root], denoted by #math.equation(block: false, alt: "the cube root of n")[$root(3, n)$]. We define the cube root as follows.


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


        #math.equation(block: true, alt: "b is a cube root of n if b cubed equals n .")[$b " " " " " is a cube root of " " " " " " " n " " " " " " " if " " " " " " " b " " " cubed equals " " " " " n .$]
        In symbols, we write


        #math.equation(block: true, alt: "b equals the cube root of n if b cubed equals n")[$bold(italic(b)) = root(3, bold(italic(n))) " " " " " if " " " " " bold(italic(b))^(3) = bold(italic(n))$]
]


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Although we cannot take the square root of a #emph[negative number], we can take the #emph[cube root] of #emph[any] real number. For example,


        #math.equation(block: true, alt: "the cube root of 64 equals 4 because 4 cubed equals 64")[$root(3, 64) = 4 " " " " " because " " " " " 4^(3) = 64$]
        and


        #math.equation(block: true, alt: "the cube root of minus 27 equals minus 3 because open parenthesis minus 3 close parenthesis cubed equals minus 27")[$root(3, − 27) = − 3 " " " " " because " " " " " ( − 3 )^(3) = − 27$]
]


      In the order of operations, simplifying radicals and powers comes after parentheses but before products and quotients.


      #examplebox("Example 3")[][
Simplify each expression.


          + #math.equation(block: false, alt: "3 the cube root of minus 8")[$3 root(3, − 8)$]
+ #math.equation(block: false, alt: "2 minus the cube root of minus 125")[$2 − root(3, − 125)$]
+ #math.equation(block: false, alt: "the fraction 6 minus the cube root of minus 27 over 2")[$display(frac(6 − root(3, − 27), 2))$]


        
        #solutionbox[
+ #math.equation(block: false, alt: "3 the cube root of minus 8 equals 3 open parenthesis minus 2 close parenthesis equals minus 6")[$3 root(3, − 8) = 3 ( − 2 ) = − 6$]
+ #math.equation(block: false, alt: "2 minus the cube root of minus 125 equals 2 minus open parenthesis minus 5 close parenthesis equals 7")[$2 − root(3, − 125) = 2 − ( − 5 ) = 7$]
+ #math.equation(block: false, alt: "the fraction 6 minus the cube root of minus 27 over 2 equals the fraction 6 minus open parenthesis minus 3 close parenthesis over 2 equals the fraction 9 over 2")[$display(frac(6 − root(3, − 27), 2) = frac(6 − ( − 3 ), 2) = frac(9, 2))$]
]
]


    
    
      
=== Scientific Notation


      Scientists and engineers regularly encounter very large numbers such as


      #math.equation(block: true, alt: "5 , 980 , 000 , 000 , 000 , 000 , 000 , 000 , 000")[$5 , 980 , 000 , 000 , 000 , 000 , 000 , 000 , 000$]
      (the mass of the Earth in kilograms) and very small numbers such as


      #math.equation(block: true, alt: "0.00000000000000000000000167")[$0.000 #h(0.167em) 000 #h(0.167em) 000 #h(0.167em) 000 #h(0.167em) 000 #h(0.167em) 000 #h(0.167em) 000 #h(0.167em) 001 #h(0.167em) 67$]
      (the mass of a hydrogen atom in grams). These numbers can be written in a more compact and useful form by using powers of #math.equation(block: false, alt: "10")[$10$].


      In our base #math.equation(block: false, alt: "10")[$10$] number system, multiplying a number by a positive power of #math.equation(block: false, alt: "10")[$10$] has the effect of moving the decimal place #math.equation(block: false, alt: "k")[$k$] places to the right, where #math.equation(block: false, alt: "k")[$k$] is the exponent in the power of #math.equation(block: false, alt: "10")[$10$]. For example,


      #math.equation(block: true, alt: "3.529 times 10 squared equals 352.9 and 25 times 10 to the power 4 equals 250 , 000")[$3 . 529 × 10^(2) = 352 . 9 " " " " " " " and " " " " " " " 25 × 10^(4) = 250 , 000$]
      Multiplying by a power of #math.equation(block: false, alt: "10")[$10$] with a negative exponent moves the decimal place to the left. For example,


      #math.equation(block: true, alt: "1728 times 10 to the power minus 3 equals 1.728 and 4.6 times 10 to the power minus 5 equals 0.000046")[$1728 × 10^(− 3) = 1.728 " " " " " " " and " " " " " " " 4.6 × 10^(− 5) = 0.000046$]
      Using this property, we can write any number as the product of a number between #math.equation(block: false, alt: "1")[$1$] and #math.equation(block: false, alt: "10")[$10$] (including #math.equation(block: false, alt: "1")[$1$]) and a power of #math.equation(block: false, alt: "10")[$10$]. For example, the mass of the Earth and the mass of a hydrogen atom can be expressed as


      #math.equation(block: true, alt: "5.98 times 1024 kilograms and 1.67 times 10 to the power minus 24 gram")[$5.98 × 1024 " kilograms" #hide($b l a n k$) " and " #hide($b l a n k$) 1.67 × 10^(− 24) " gram"$]
      respectively. A number written in this form is said to be expressed in #strong[scientific notation].


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[To Write a Number in Scientific Notation:]


        + Locate the decimal point so that there is exactly one nonzero digit to its left.
+ Count the number of places you moved the decimal point: This determines the power of #math.equation(block: false, alt: "10")[$10$].
            + If the original number is greater than #math.equation(block: false, alt: "10")[$10$], the exponent is positive.
+ If the original number is less than #math.equation(block: false, alt: "1")[$1$], the exponent is negative.
]


      #examplebox("Example 4")[][
Write each number in scientific notation.


        + #math.equation(block: false, alt: "478 , 000, equals 4.78000 times 10 to the power 5, Move the decimal 5 places.; equals 4.78 times 10 to the power 5")[$478 , 000 & = 4.78000 × 10^(5) &  & "Move the decimal 5 places." \  & = 4.78 × 10^(5)$]
+ #math.equation(block: false, alt: "0.00032, equals 00003.2 times 10 to the power minus 4, Move the decimal 4 places.; equals 3.2 times 10 to the power minus 4")[$0.00032 & = 00003.2 × 10^(− 4) &  & "Move the decimal 4 places." \  & = 3.2 × 10^(− 4)$]
]


      #examplebox("Example 5")[][
The average American eats #math.equation(block: false, alt: "110")[$110$] kilograms of meat per year. It takes about #math.equation(block: false, alt: "16")[$16$] kilograms of grain to produce #math.equation(block: false, alt: "1")[$1$] kilogram of meat, and advanced farming techniques can produce about #math.equation(block: false, alt: "6000")[$6000$] kilograms of grain on each hectare of arable land. (The hectare is #math.equation(block: false, alt: "10 , 000")[$10 , 000$] square meters, or just under #math.equation(block: false, alt: "2 the fraction 1 over 2")[$2 frac(1, 2)$] acres.) Now, the total land area of the Earth is about #math.equation(block: false, alt: "13")[$13$] billion hectares, but only about #math.equation(block: false, alt: "11 %")[$11 upright(%)$] of that land is arable. Is it possible for each of the #math.equation(block: false, alt: "7.6")[$7.6$] billion people on Earth to eat as much meat as Americans do?


        
        #solutionbox[
First we will compute the amount of meat necessary to feed every person on Earth #math.equation(block: false, alt: "110")[$110$] kilograms per year. In 2018 there are #math.equation(block: false, alt: "7.6 times 10 to the power 9")[$7.6 × 10^(9)$] people on Earth.


          #math.equation(block: true, alt: "open parenthesis 5.5 times 10 to the power 9 people close parenthesis times open parenthesis 110 kg/person close parenthesis equals 8.36 times 10 to the power 11 kg of meat")[$( 5.5 × 10^(9) " people" ) × ( 110 " kg/person" ) = 8.36 × 10^(11) " kg of meat"$]
          Next we will compute the amount of grain needed to produce that much meat.


          #math.equation(block: true, alt: "open parenthesis 16 kg of grain/kg of meat close parenthesis times open parenthesis 8.36 times 10 to the power 11 kg of meat close parenthesis equals 1.34 times 10 to the power 13 kg of grain")[$( 16 " kg of grain/kg of meat" ) × ( 8.36 × 10^(11) " kg of meat" ) = 1.34 × 10^(13) " kg of grain"$]
          Next we will see how many hectares of land are needed to produce that much grain.


          #math.equation(block: true, alt: "open parenthesis 1.34 times 10 to the power 13 kg of grain close parenthesis divided by open parenthesis 6000 kg/hectare close parenthesis equals 2.23 times 10 to the power 11 hectares")[$( 1.34 × 10^(13) " kg of grain" ) ÷ ( 6000 " kg/hectare" ) = 2.23 × 10^(11) " hectares"$]
          Finally, we will compute the amount of arable land available for grain production.


          #math.equation(block: true, alt: "0.11 times open parenthesis 13 times 10 to the power 9 hectares close parenthesis equals 1.43 times 10 to the power 9 hectares")[$0.11 × ( 13 × 10^(9) " hectares" ) = 1.43 × 10^(9) " hectares"$]
          Thus, even if we use every hectare of arable land to produce grain for livestock, we will not have enough to provide every person on Earth with #math.equation(block: false, alt: "110")[$110$] kilograms of meat per year.
]
]


    
    
      
=== Section Summary


      
        
==== Vocabulary


        Look up the definitions of new terms in the Glossary.


        - Order of operations
- Scientific notation
- Radical
- Cube root
- Grouping symbol
- Fraction bar
- Square root bar


      
      
        
==== SKILLS


        Practice each skill in the exercises listed.


        + Follow the order of operations: \#1–26
+ Compute cube roots: \#27–39
+ Use a calculator to simplify expressions: \#31–42
+ Evaluate an expression: \#43–50
+ Convert between standard and scientific notation: \#51–54
+ Compute using scientific notation: \#55–62


      
    
    
      
=== Exercises A.1


      For Problems 1-26, simplify each expression according to the order of operations.


      
        #math.equation(block: true, alt: "the fraction 3 open parenthesis 6 minus 8 close parenthesis over minus 2 minus the fraction 6 over minus 2")[$display(frac(3 ( 6 − 8 ), − 2)) − display(frac(6, − 2))$]


        
          #math.equation(block: true, alt: "6")[$6$]


        
      
      
        #math.equation(block: true, alt: "the fraction 5 open parenthesis 3 minus 5 close parenthesis over 2 minus the fraction 18 over minus 3")[$display(frac(5 ( 3 − 5 ), 2)) − display(frac(18, − 3))$]


      
      
        #math.equation(block: true, alt: "6 open bracket 3 minus 2 open parenthesis 4 plus 1 close parenthesis close bracket")[$6 [ 3 − 2 ( 4 + 1 ) ]$]


        
          #math.equation(block: true, alt: "minus 42")[$− 42$]


        
      
      
        #math.equation(block: true, alt: "5 open bracket 3 plus 4 open parenthesis 6 minus 4 close parenthesis close bracket")[$5 [ 3 + 4 ( 6 − 4 ) ]$]


      
      
        #math.equation(block: true, alt: "open parenthesis 4 minus 3 close parenthesis open bracket 2 plus 3 open parenthesis 2 minus 1 close parenthesis close bracket")[$( 4 − 3 ) [ 2 + 3 ( 2 − 1 ) ]$]


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


        
      
      
        #math.equation(block: true, alt: "open parenthesis 8 minus 6 close parenthesis open bracket 5 plus 7 open parenthesis 2 minus 3 close parenthesis close bracket")[$( 8 − 6 ) [ 5 + 7 ( 2 − 3 ) ]$]


        
          #math.equation(block: true, alt: "6")[$6$]


        
      
      
        #math.equation(block: true, alt: "64 divided by 8 open bracket 4 minus 2 open parenthesis 3 plus 1 close parenthesis close bracket")[$64 ÷ 8 [ 4 − 2 ( 3 + 1 ) ]$]


        
          #math.equation(block: true, alt: "minus 2")[$− 2$]


        
      
      
        #math.equation(block: true, alt: "27 divided by 3 open bracket 9 minus 3 open parenthesis 4 minus 2 close parenthesis close bracket")[$27 ÷ 3 [ 9 − 3 ( 4 − 2 ) ]$]


      
      
        #math.equation(block: true, alt: "5 open bracket 3 plus open parenthesis 8 minus 1 close parenthesis close bracket divided by open parenthesis minus 25 close parenthesis")[$5 [ 3 + ( 8 − 1 ) ] ÷ ( − 25 )$]


        
          #math.equation(block: true, alt: "minus 2")[$− 2$]


        
      
      
        #math.equation(block: true, alt: "minus 3 open bracket minus 2 plus open parenthesis 6 minus 1 close parenthesis close bracket divided by 9")[$− 3 [ − 2 + ( 6 − 1 ) ] ÷ 9$]


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


        
      
      
        #math.equation(block: true, alt: "open bracket minus 3 open parenthesis 8 minus 2 close parenthesis plus 3 close bracket times open bracket 24 divided by 6 close bracket")[$[ − 3 ( 8 − 2 ) + 3 ] ⋅ [ 24 ÷ 6 ]$]


        
          #math.equation(block: true, alt: "minus 60")[$− 60$]


        
      
      
        #math.equation(block: true, alt: "open bracket minus 2 plus 3 open parenthesis 5 minus 8 close parenthesis close bracket times open bracket minus 15 divided by 3 close bracket")[$[ − 2 + 3 ( 5 − 8 ) ] ⋅ [ − 15 ÷ 3 ]$]


      
      
        #math.equation(block: true, alt: "minus 5 squared")[$− 5^(2)$]


        
          #math.equation(block: true, alt: "minus 25")[$− 25$]


        
      
      
        #math.equation(block: true, alt: "open parenthesis minus 15 close parenthesis squared")[$( − 15 )^(2)$]


      
      
        #math.equation(block: true, alt: "open parenthesis minus 3 close parenthesis to the power 4")[$( − 3 )^(4)$]


        
          #math.equation(block: true, alt: "81")[$81$]


        
      
      
        #math.equation(block: true, alt: "minus 3 to the power 4")[$− 3^(4)$]


      
      
        #math.equation(block: true, alt: "minus 4 cubed")[$− 4^(3)$]


        
          #math.equation(block: true, alt: "minus 64")[$− 64$]


        
      
      
        #math.equation(block: true, alt: "open parenthesis minus 4 close parenthesis cubed")[$( − 4 )^(3)$]


      
      
        #math.equation(block: true, alt: "open parenthesis minus 2 close parenthesis to the power 5")[$( − 2 )^(5)$]


        
          #math.equation(block: true, alt: "minus 32")[$− 32$]


        
      
      
        #math.equation(block: true, alt: "minus 2 to the power 5")[$− 2^(5)$]


      
      
        #math.equation(block: true, alt: "the fraction 4 times 2 cubed over 16 plus 3 times 4 squared")[$display(frac(4 ⋅ 2^(3), 16)) + 3 ⋅ 4^(2)$]


        
          #math.equation(block: true, alt: "50")[$50$]


        
      
      
        #math.equation(block: true, alt: "the fraction 4 times 3 squared over 6 plus open parenthesis 3 times 4 close parenthesis squared")[$display(frac(4 ⋅ 3^(2), 6)) + ( 3 ⋅ 4 )^(2)$]


      
      
        #math.equation(block: true, alt: "the fraction 3 squared minus 5 over 6 minus 2 squared minus the fraction 6 squared over 3 squared")[$display(frac(3^(2) − 5, 6 − 2^(2))) − display(frac(6^(2), 3^(2)))$]


        
          #math.equation(block: true, alt: "minus 2")[$− 2$]


        
      
      
        #math.equation(block: true, alt: "the fraction 3 times 2 squared over 4 minus 1 plus the fraction open parenthesis minus 3 close parenthesis open parenthesis 2 close parenthesis cubed over 6")[$display(frac(3 ⋅ 2^(2), 4 − 1)) + display(frac(( − 3 ) ( 2 )^(3), 6))$]


      
      
        #math.equation(block: true, alt: "the fraction open parenthesis minus 5 close parenthesis squared minus 3 squared over 4 minus 6 plus the fraction open parenthesis minus 3 close parenthesis squared over 2 plus 1")[$display(frac(( − 5 )^(2) − 3^(2), 4 − 6)) + display(frac(( − 3 )^(2), 2 + 1))$]


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


        
      
      
        #math.equation(block: true, alt: "the fraction 7 squared minus 6 squared over 10 plus 3 minus the fraction 8 squared times open parenthesis minus 2 close parenthesis over open parenthesis minus 4 close parenthesis squared")[$display(frac(7^(2) − 6^(2), 10 + 3)) − display(frac(8^(2) ⋅ ( − 2 ), ( − 4 )^(2)))$]


      
      For Problems 27-28, compute each cube root. Round your answers to three decimal places if necessary. Verify your answers by cubing them.


      
        + #math.equation(block: false, alt: "the cube root of 512")[$root(3, 512)$]
+ #math.equation(block: false, alt: "the cube root of minus 125")[$root(3, − 125)$]
+ #math.equation(block: false, alt: "the cube root of minus 0.064")[$root(3, − 0.064)$]
+ #math.equation(block: false, alt: "the cube root of 1.728")[$root(3, 1.728)$]


        
          + #math.equation(block: false, alt: "8")[$8$]
+ #math.equation(block: false, alt: "minus 5")[$− 5$]
+ #math.equation(block: false, alt: "0.4")[$0.4$]
+ #math.equation(block: false, alt: "1.2")[$1.2$]


        
      
      
        + #math.equation(block: false, alt: "the cube root of 9")[$root(3, 9)$]
+ #math.equation(block: false, alt: "the cube root of 258")[$root(3, 258)$]
+ #math.equation(block: false, alt: "the cube root of minus 0.002")[$root(3, − 0.002)$]
+ #math.equation(block: false, alt: "the cube root of minus 3.1")[$root(3, − 3.1)$]


      
      For Problems 29-30, simplify each expression according to the order of operations.


      
        + #math.equation(block: false, alt: "the fraction 4 minus 3 the cube root of 64 over 2")[$display(frac(4 − 3 root(3, 64), 2))$]
+ #math.equation(block: false, alt: "the fraction 4 plus the cube root of minus 216 over 8 minus 8 the cube root of 8")[$display(frac(4 + root(3, − 216), 8 − 8 root(3, 8)))$]


        
          + #math.equation(block: false, alt: "minus 4")[$− 4$]
+ #math.equation(block: false, alt: "the fraction minus 1 over 3")[$display(frac(− 1, 3))$]


        
      
      
        + #math.equation(block: false, alt: "the cube root of 3 cubed plus 4 cubed plus 5 cubed")[$root(3, 3^(3) + 4^(3) + 5^(3))$]
+ #math.equation(block: false, alt: "the cube root of 9 cubed plus 10 cubed minus 1 cubed")[$root(3, 9^(3) + 10^(3) − 1^(3))$]


      
      For Problems 31-42, use a calculator to simplify each expression.


      
        #math.equation(block: true, alt: "the fraction minus 8398 over 26 times 17")[$display(frac(− 8398, 26 ⋅ 17))$]


        
          #math.equation(block: true, alt: "minus 19")[$− 19$]


        
      
      
        #math.equation(block: true, alt: "the fraction minus 415.112 over 8.58 plus 18.73")[$display(frac(− 415.112, 8.58 + 18.73))$]


      
      
        #math.equation(block: true, alt: "the fraction 112.78 plus 2599.124 over 27.56")[$display(frac(112.78 + 2599.124, 27.56))$]


        
          #math.equation(block: true, alt: "98.4")[$98.4$]


        
      
      
        #math.equation(block: true, alt: "the fraction 202 , 462 minus 9510 over 356")[$display(frac(202 "," 462 − 9510, 356))$]


      
      
        #math.equation(block: true, alt: "the square root of 24 times 54")[$sqrt(24 ⋅ 54)$]


        
          #math.equation(block: true, alt: "36")[$36$]


        
      
      
        #math.equation(block: true, alt: "the square root of the fraction 1216 over 19")[$sqrt(display(frac(1216, 19)))$]


      
      
        #math.equation(block: true, alt: "the fraction 116 minus 35 over 215 minus 242")[$display(frac(116 − 35, 215 − 242))$]


        
          #math.equation(block: true, alt: "minus 3")[$− 3$]


        
      
      
        #math.equation(block: true, alt: "the fraction 842 minus 987 over 443 minus 385")[$display(frac(842 − 987, 443 − 385))$]


      
      
        #math.equation(block: true, alt: "the square root of 27 squared plus 36 squared")[$sqrt(27^(2) + 36^(2))$]


        
          #math.equation(block: true, alt: "45")[$45$]


        
      
      
        #math.equation(block: true, alt: "the square root of 13 squared minus 4 times 21 times 2")[$sqrt(13^(2) − 4 ⋅ 21 ⋅ 2)$]


      
      
        #math.equation(block: true, alt: "the fraction minus 27 minus the square root of 27 squared minus 4 open parenthesis 4 close parenthesis open parenthesis 35 close parenthesis over 2 times 4")[$display(frac(− 27 − sqrt(27^(2) − 4 ( 4 ) ( 35 )), 2 ⋅ 4))$]


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


        
      
      
        #math.equation(block: true, alt: "the fraction 13 plus the square root of 13 squared minus 4 open parenthesis 5 close parenthesis open parenthesis minus 6 close parenthesis over 2 times 5")[$display(frac(13 + sqrt(13^(2) − 4 ( 5 ) ( − 6 )), 2 ⋅ 5))$]


      
      For Problems 43-50, evaluate the expression for the given values of the variable. Use your calculator where appropriate.


      
        #math.equation(block: true, alt: "the fraction 5 open parenthesis F minus 32 close parenthesis over 9")[$display(frac(5 ( F − 32 ), 9))$]; #math.equation(block: true, alt: "F equals 212")[$" " " " " " F = 212$]


        
          #math.equation(block: true, alt: "100")[$100$]


        
      
      
        #math.equation(block: false, alt: "the fraction a minus 4 s over 1 minus r")[$display(frac(a − 4 s, 1 − r))$]; #math.equation(block: false, alt: "r equals 2 , s equals 12 ,")[$" " " " " " r = 2 , " " s = 12 , " "$] and #math.equation(block: false, alt: "a equals 4")[$" " a = 4$]


      
      
        #math.equation(block: false, alt: "P plus P r t")[$P + P r t$]; #math.equation(block: false, alt: "P equals 1000 , r equals 0.04 ,")[$" " " " " " P = 1000 , " " r = 0.04 , " "$] and #math.equation(block: false, alt: "t equals 2")[$" " t = 2$]


        
          #math.equation(block: true, alt: "1080")[$1080$]


        
      
      
        #math.equation(block: false, alt: "R open parenthesis 1 plus a t close parenthesis")[$R ( 1 + a t )$]; #math.equation(block: false, alt: "R equals 2.5 , a equals 0.05 ,")[$" " " " " " R = 2.5 , " " a = 0.05 , " "$] and #math.equation(block: false, alt: "t equals 20")[$" " t = 20$]


      
      
        #math.equation(block: false, alt: "the fraction 1 over 2 g t squared minus 12 t")[$display(frac(1, 2)) g t^(2) − 12 t$]; #math.equation(block: false, alt: "g equals 32")[$" " " " " " g = 32 " "$] and #math.equation(block: false, alt: "t equals the fraction 3 over 4")[$" " t = display(frac(3, 4))$]


        
          #math.equation(block: true, alt: "0")[$0$]


        
      
      
        #math.equation(block: false, alt: "the fraction M v squared over g")[$display(frac(M v^(2), g))$]; #math.equation(block: false, alt: "M equals the fraction 16 over 3 , a equals the fraction 3 over 2 ,")[$" " " " " " M = display(frac(16, 3)) , " " a = display(frac(3, 2)) , " "$] and #math.equation(block: false, alt: "g equals 32")[$" " g = 32$]


      
      
        #math.equation(block: false, alt: "the fraction 32 open parenthesis V minus v close parenthesis squared over g")[$display(frac(32 ( V − v )^(2), g))$]; #math.equation(block: false, alt: "V equals 12.78 , v equals 4.26 ,")[$" " " " " " V = 12.78 , " " v = 4.26 , " "$] and #math.equation(block: false, alt: "g equals 32")[$" " g = 32$]


        
          #math.equation(block: true, alt: "72.5904")[$72.5904$]


        
      
      
        #math.equation(block: false, alt: "the fraction 32 open parenthesis V minus v close parenthesis squared over g")[$display(frac(32 ( V − v )^(2), g))$]; #math.equation(block: false, alt: "V equals 38.3 , v equals minus 6.7 ,")[$" " " " " " V = 38.3 , " " v = − 6.7 , " "$] and #math.equation(block: false, alt: "g equals 9.8")[$" " g = 9.8$]


      
      For Problems 51-52, write each number in scientific notation.


      
        + #math.equation(block: false, alt: "285")[$285$]
+ #math.equation(block: false, alt: "8 , 372 , 000")[$8 , 372 , 000$]
+ #math.equation(block: false, alt: "0.024")[$0.024$]
+ #math.equation(block: false, alt: "0.000523")[$0.000523$]


        
          + #math.equation(block: false, alt: "2.85 times 10 squared")[$2.85 × 10^(2)$]
+ #math.equation(block: false, alt: "8.372 times 10 to the power 6")[$8.372 × 10^(6)$]
+ #math.equation(block: false, alt: "2.4 times 10 to the power minus 2")[$2.4 × 10^(− 2)$]
+ #math.equation(block: false, alt: "5.23 times 10 to the power minus 4")[$5.23 × 10^(− 4)$]


        
      
      
        + #math.equation(block: false, alt: "68 , 742")[$68 , 742$]
+ #math.equation(block: false, alt: "481 , 000 , 000 , 000")[$481 , 000 , 000 , 000$]
+ #math.equation(block: false, alt: "0.421")[$0.421$]
+ #math.equation(block: false, alt: "0.000004")[$0.000004$]


      
      For Problems 53-54, write each number in standard notation.


      
        + #math.equation(block: false, alt: "2.4 times 10 squared")[$2.4 × 10^(2)$]
+ #math.equation(block: false, alt: "6.87 times 10 to the power 15")[$6.87 × 10^(15)$]
+ #math.equation(block: false, alt: "5.0 times 10 to the power minus 3")[$5.0 × 10^(− 3)$]
+ #math.equation(block: false, alt: "2.02 times 10 to the power minus 4")[$2.02 × 10^(− 4)$]


        
          + #math.equation(block: false, alt: "240")[$240$]
+ #math.equation(block: false, alt: "6 , 870 , 000 , 000 , 000 , 000")[$6 , 870 , 000 , 000 , 000 , 000$]
+ #math.equation(block: false, alt: "0.005")[$0.005$]
+ #math.equation(block: false, alt: "0.000202")[$0.000202$]


        
      
      
        + #math.equation(block: false, alt: "4.8 times 10 cubed")[$4.8 × 10^(3)$]
+ #math.equation(block: false, alt: "8.31 times 10 to the power 12")[$8.31 × 10^(12)$]
+ #math.equation(block: false, alt: "8.0 times 10 to the power minus 1")[$8.0 × 10^(− 1)$]
+ #math.equation(block: false, alt: "4.31 times 10 to the power minus 5")[$4.31 × 10^(− 5)$]


      
      For Problems 55-56, compute with the aid of a calculator. Write your answers in standard notation.


      
        + #math.equation(block: false, alt: "the fraction open parenthesis 2.4 times 10 to the power minus 8 close parenthesis open parenthesis 6.5 times 10 to the power 32 close parenthesis over 5.2 times 10 to the power 18")[$display(frac(( 2.4 × 10^(− 8) ) ( 6.5 × 10^(32) ), 5.2 × 10^(18)))$]
+ #math.equation(block: false, alt: "the fraction open parenthesis 7.5 times 10 to the power minus 13 close parenthesis open parenthesis 3.6 times 10 to the power minus 9 close parenthesis over open parenthesis 1.5 times 10 to the power minus 15 close parenthesis open parenthesis 1.6 times 10 to the power minus 11 close parenthesis")[$display(frac(( 7.5 × 10^(− 13) ) ( 3.6 × 10^(− 9) ), ( 1.5 × 10^(− 15) ) ( 1.6 × 10^(− 11) )))$]


        
          + #math.equation(block: false, alt: "3 , 000 , 000")[$3 , 000 , 000$]
+ #math.equation(block: false, alt: "112 , 500")[$112 , 500$]


        
      
      
        + #math.equation(block: false, alt: "the fraction open parenthesis 8.4 times 10 to the power minus 22 close parenthesis open parenthesis 1.6 times 10 to the power 15 close parenthesis over 3.2 times 10 to the power minus 11")[$display(frac(( 8.4 × 10^(− 22) ) ( 1.6 × 10^(15) ), 3.2 × 10^(− 11)))$]
+ #math.equation(block: false, alt: "the fraction open parenthesis 9.4 times 10 to the power 24 close parenthesis open parenthesis 7.2 times 10 to the power minus 18 close parenthesis over open parenthesis 4.5 times 10 to the power 26 close parenthesis open parenthesis 6.4 times 10 to the power minus 16 close parenthesis")[$display(frac(( 9.4 × 10^(24) ) ( 7.2 × 10^(− 18) ), ( 4.5 × 10^(26) ) ( 6.4 × 10^(− 16) )))$]


      
      
        In 2018, the public debt of the United States was over \$20,620,000,000,000.


        + Express this number in scientific notation.
+ If the population of the United States in 2018 was 327,112,000, what was the per capita debt (the debt per person) in 2018?


        
          + #math.equation(block: false, alt: "2.062 times 10 to the power 13")[$2.062 × 10^(13)$]
+ \$63,036.51


        
      
      
        A light-year is the number of miles traveled by light in 1 year (365 days). The speed of light is approximately 186,000 miles per second.


        + Compute the number of miles in 1 light-year, and express your answer in scientific notation.
+ The star nearest to the Sun is Proxima Centauri, at a distance of 4.3 light-hears. How long would it take #strong[Pioneer 10] (the first space vehicle to achieve escape velocity from the solar system), traveling at 32,114 miles per hour, to reach Proxima Centauri?


      
      
        The diameter of the galactic disk is about #math.equation(block: false, alt: "1.2 times 10 to the power 18")[$1.2 × 10^(18)$] kilometers, and our Sun lies about halfway from the center of the galaxy to the edge of the disk. The Sun orbits the galactic center once in 240 million years.


        + What is the speed of the Sun in its orbit, in kilometers per year?
+ What is its speed in meters per second?


        
          + #math.equation(block: false, alt: "7.9 times 10 to the power 9")[$7.9 × 10^(9)$] km per year
+ 250,000 meters per second


        
      
      
        Lake Superior has an area of 31,700 square miles and an average depth of 483 feet.


        + Find the approximate volume of Lake Superior in cubic feet.
+ If 1 cubic foot of water is equivalent to 7.48 gallons, how many gallons of water are in Lake Superior?


      
      
        The average distance from the Earth to the Sun is #math.equation(block: false, alt: "1.5 times 10 to the power 11")[$1.5 × 10^(11)$] meters. The distance from the Sun to Proxima Centauri, the next closest star, is #math.equation(block: false, alt: "3.99 times 10 to the power 16")[$3.99 × 10^(16)$] meters. The most distant star visible to the unaided eye are 2000 times as far away as Proxima Centauri.


        + How many times farther is Proxima Centauri from the Sun than the Sun is from Earth?
+ How far from the Sun are the most distant visible stars?


        
          + 250,000 times
+ #math.equation(block: false, alt: "8 times 10 to the power 19")[$8 × 10^(19)$] meters


        
      
      
        The radius of the Earth is #math.equation(block: false, alt: "6.37 times 10 to the power 6")[$6.37 × 10^(6)$] meters, and the radius of the Sun is #math.equation(block: false, alt: "6.96 times 10 to the power 8")[$6.96 × 10^(8)$] meters. The radii of the other stars range from 1% of the solar radius to 1000 times the solar radius.


        + What fraction of the solar radius is the Earth's radius?
+ What is the range of stellar radii, in meters?