Chapter 7 Review and Practice Test

Learning objectives

Learning objectives

• Review the key ideas of this chapter.

• Apply the chapter's methods to review exercises.

• Check your understanding with a practice test.

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Worked example

Additive Identity

additive identity

Additive Inverse

additive inverse

additive inverse

Irrational number

irrational number

Multiplicative Identity

multiplicative identity

Multiplicative Inverse

multiplicative inverse

multiplicative inverse

a

Rational number

rational number

p

q

Real number

real number

##

Key Concepts

• Real numbers

• Commutative PropertiesCommutative Property of Addition:If $a,b$ are real numbers, then $a+b=b+a$ Commutative Property of Multiplication:If $a,b$ are real numbers, then $a\cdot{}b=b\cdot{}a$

• Associative PropertiesAssociative Property of Addition:If $a,b,c$ are real numbers then $(a+b)+c=a+(b+c)$ Associative Property of Multiplication:If $a,b,c$ are real numbers then $(a\cdot{}b)\cdot{}c=a\cdot(b\cdot{}c)$

• Distributive Property:If $a,b,c$ are real numbers then$a(b+c)=ab+ac$$(b+c)a=ba+ca$$a(b-c)=ab-ac$

• Identity PropertiesIdentity Property of Addition: For any real number a: $a+0=a\,0+a=a\,$ 0 is the additive identity Identity Property of Multiplication: For any real number a: $a\cdot{}1=a\,1\cdot{}a=a\,$ 1 is the multiplicative identity

• Inverse PropertiesInverse Property of Addition: For any real number a: $a+(-a)=0\,-a$ is the additive inverse of a Inverse Property of Multiplication: For any real number a: $(a\ne{}0)\,a\cdot\frac{1}{a}=1\,\frac{1}{a}$ is the multiplicative inverse of a

• Properties of ZeroMultiplication by Zero: For any real number a, $\begin{array}{ccccccc} a\cdot{}0=0 & & & 0\cdot{}a=0 & & & \mathrm{The product of any number and 0 is 0.} \\ \end{array}$ Division of Zero: For any real number a, $\begin{array}{ccccccc} \frac{0}{a}=0 & & & & & & \mathrm{Zero divided by any real number, except itself, is zero.} \\ \end{array}$ Division by Zero:For any real number a, $\frac{a}{0}$ is undefined and $a\mathrm{÷}0$ is undefined. Division by zero is undefined.

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Review Exercises

In the following exercises, write as the ratio of two integers.

In the following exercises, determine which of the numbers is rational.

In the following exercises, identify whether each given number is rational or irrational.

In the following exercises, list the ⓐ whole numbers, ⓑ integers, ⓒ rational numbers, ⓓ irrational numbers, ⓔ real numbers for each set of numbers.

In the following exercises, use the commutative property to rewrite the given expression.

In the following exercises, use the associative property to rewrite the given expression.

In the following exercises, evaluate each expression for the given value.

In the following exercises, simplify using the commutative and associative properties.

In the following exercises, simplify using the distributive property.

In the following exercises, evaluate using the distributive property.

In the following exercises, identify whether each example is using the identity property of addition or multiplication.

In the following exercises, find the additive inverse.

In the following exercises, find the multiplicative inverse.

In the following exercises, simplify.

In the following exercises, convert between U.S. units. Round to the nearest tenth.

In the following exercises, solve, and state your answer in mixed units.

In the following exercises, convert between metric units.

In the following exercises, solve.

In the following exercises, convert between U.S. and metric units. Round to the nearest tenth.

In the following exercises, convert the Fahrenheit temperatures to degrees Celsius. Round to the nearest tenth.

In the following exercises, convert the Celsius temperatures to degrees Fahrenheit. Round to the nearest tenth.

##

In the following exercises, simplify the given expression.

In the following exercises, solve using the appropriate unit conversions.

Key Terms

Additive Identity

additive identity

Additive Inverse

additive inverse

additive inverse

Irrational number

irrational number

Multiplicative Identity

multiplicative identity

Multiplicative Inverse

multiplicative inverse

multiplicative inverse

a

Rational number

rational number

p

q

Real number

real number

##

Key Concepts

• Real numbers

• Commutative PropertiesCommutative Property of Addition:If $a,b$ are real numbers, then $a+b=b+a$ Commutative Property of Multiplication:If $a,b$ are real numbers, then $a\cdot{}b=b\cdot{}a$

• Associative PropertiesAssociative Property of Addition:If $a,b,c$ are real numbers then $(a+b)+c=a+(b+c)$ Associative Property of Multiplication:If $a,b,c$ are real numbers then $(a\cdot{}b)\cdot{}c=a\cdot(b\cdot{}c)$

• Distributive Property:If $a,b,c$ are real numbers then$a(b+c)=ab+ac$$(b+c)a=ba+ca$$a(b-c)=ab-ac$

• Identity PropertiesIdentity Property of Addition: For any real number a: $a+0=a\,0+a=a\,$ 0 is the additive identity Identity Property of Multiplication: For any real number a: $a\cdot{}1=a\,1\cdot{}a=a\,$ 1 is the multiplicative identity

• Inverse PropertiesInverse Property of Addition: For any real number a: $a+(-a)=0\,-a$ is the additive inverse of a Inverse Property of Multiplication: For any real number a: $(a\ne{}0)\,a\cdot\frac{1}{a}=1\,\frac{1}{a}$ is the multiplicative inverse of a

• Properties of ZeroMultiplication by Zero: For any real number a, $\begin{array}{ccccccc} a\cdot{}0=0 & & & 0\cdot{}a=0 & & & \mathrm{The product of any number and 0 is 0.} \\ \end{array}$ Division of Zero: For any real number a, $\begin{array}{ccccccc} \frac{0}{a}=0 & & & & & & \mathrm{Zero divided by any real number, except itself, is zero.} \\ \end{array}$ Division by Zero:For any real number a, $\frac{a}{0}$ is undefined and $a\mathrm{÷}0$ is undefined. Division by zero is undefined.

##

Review Exercises

In the following exercises, write as the ratio of two integers.

In the following exercises, determine which of the numbers is rational.

In the following exercises, identify whether each given number is rational or irrational.

In the following exercises, list the ⓐ whole numbers, ⓑ integers, ⓒ rational numbers, ⓓ irrational numbers, ⓔ real numbers for each set of numbers.

In the following exercises, use the commutative property to rewrite the given expression.

In the following exercises, use the associative property to rewrite the given expression.

In the following exercises, evaluate each expression for the given value.

In the following exercises, simplify using the commutative and associative properties.

In the following exercises, simplify using the distributive property.

In the following exercises, evaluate using the distributive property.

In the following exercises, identify whether each example is using the identity property of addition or multiplication.

In the following exercises, find the additive inverse.

In the following exercises, find the multiplicative inverse.

In the following exercises, simplify.

In the following exercises, convert between U.S. units. Round to the nearest tenth.

In the following exercises, solve, and state your answer in mixed units.

In the following exercises, convert between metric units.

In the following exercises, solve.

In the following exercises, convert between U.S. and metric units. Round to the nearest tenth.

In the following exercises, convert the Fahrenheit temperatures to degrees Celsius. Round to the nearest tenth.

In the following exercises, convert the Celsius temperatures to degrees Fahrenheit. Round to the nearest tenth.

##

In the following exercises, simplify the given expression.

In the following exercises, solve using the appropriate unit conversions.

1. $-5$

2. $1.8$

3. $\mathrm{0.75319…},0.\overset{---}{16},1.95$

4. ⓐ $\,\sqrt{72}$ ⓑ $\,\sqrt{64}$

1. $-5,\,-2\frac{1}{4},\,-\sqrt{4},\,0.\overset{---}{25},\,\frac{13}{5},\,4$

2. $-14\cdot{}5=\mathrm{\_\_\_\_}$

3. $a+8=\mathrm{\_\_\_\_}$

4. $(22+7)+3=\mathrm{\_\_\_\_\_}$

1. $\frac{1}{2}(22y)=\mathrm{\_\_\_\_\_}$

2. If $z=-\frac{5}{3},$ evaluate: ⓐ $\,z+5.39+(-z)$ ⓑ $\,z+(-z)+5.39$

3. If $m=-13,$ evaluate: ⓐ $\,-\frac{2}{5}(\frac{5}{2}m)$ ⓑ $\,(-\frac{2}{5}\cdot\frac{5}{2})m$

4. $\frac{1}{4}+\frac{11}{15}+(-\frac{1}{4})$

1. $-18\cdot{}15\cdot\frac{2}{9}$

2. $(3.98d+0.75d)+1.25d$

3. $30(\frac{5}{6}q)$

4. $52m+(-20n)+(-18m)+(-5n)$

1. $9(u-4)$

2. $-8(-7a-12)$

3. $(y+10)\cdot{}p$

4. $4(x+3)-8(x-7)$

1. If $n=\frac{7}{8},$ evaluate ⓐ $\,8(n+\frac{1}{4})$ and ⓑ $\,8\cdot{}n+8\cdot\frac{1}{4}$ to show that $8(n+\frac{1}{4})=8\cdot{}n+8\cdot\frac{1}{4}$

2. If $y=-18,$ evaluate ⓐ $\,-(y-18)$ and ⓑ $\,-y+18$ to show that $-(y-18)=-y+18$

3. $29+0=29$

4. $9\cdot{}1+(-3)=9+(-3)$

1. $19.4$

2. $-\frac{7}{15}$

3. $-5$

4. $-\frac{4}{9}$

1. $(n+6.75)+0.25$

2. $9(6x-11)+15$

3. A picture frame is $42$ inches wide. Convert the width to feet.

4. A playground is $45$ feet wide. Convert the width to yards.

1. Shamu weighs $4.5$ tons. Convert the weight to pounds.

2. How many tablespoons are in a quart?

3. Trinh needs $30$ cups of paint for her class art project. Convert the volume to gallons.

4. Every day last week, Pedro recorded the amount of time he spent reading. He read for $50,25,83,45,32,60,\mathrm{and}\,135$ minutes. How much time, in hours and minutes, did Pedro spend reading?

1. Dalila wants to make pillow covers. Each cover takes $30$ inches of fabric. How many yards and inches of fabric does she need for $4$ pillow covers?

2. Mount Everest is $8,850$ meters tall. Convert the height to kilometers.

3. One cup of yogurt contains $13$ grams of protein. Convert this to milligrams.

4. A bottle of water contained $650$ milliliters. Convert this to liters.

1. Selma had a $\mathrm{1-liter}$ bottle of water. If she drank $145$ milliliters, how much water, in milliliters, was left in the bottle?

2. One ounce of tofu provides $2$ grams of protein. How many milligrams of protein are provided by $5$ ounces of tofu?

3. A college basketball court is $84$ feet long. Convert this length to meters.

4. Lucas weighs $78$ kilograms. Convert his weight to pounds.

1. A box of books weighs $25$ pounds. Convert this weight to kilograms.

2. $23\mathrm{^{\circ}F}$

3. $64\mathrm{^{\circ}F}$

4. $-5\mathrm{^{\circ}C}$

1. $24\mathrm{^{\circ}C}$

2. Is $\sqrt{144}$ rational or irrational?

3. Rewrite using the commutative property: $x\cdot{}14=\,\mathrm{\_\_\_\_\_\_\_\_\_}$

4. Rewrite the expression using the associative property: $(8\cdot{}2)\cdot{}5=\,\mathrm{\_\_\_\_\_\_\_\_\_\_\_}$

1. For the number $\frac{2}{5}$ find the ⓐ additive inverse ⓑ multiplicative inverse.

2. $-3+15y+3$

3. $(\frac{8}{15}+\frac{2}{9})+\frac{7}{9}$

4. $14y+(-6z)+16y+2z$

1. $6(5x-4)$

2. $\frac{1}{4}(8a+12)$

3. $8(6p-1)+2(9p+3)$

4. One cup of milk contains $276$ milligrams of calcium. Convert this to grams. $\mathrm{(1 milligram}=\mathrm{0.001 gram)}$

1. Janice ran $15$ kilometers. Convert this distance to miles. Round to the nearest hundredth of a mile. $\mathrm{(1 mile}=\mathrm{1.61 kilometers)}$

2. Use the formula $F=\frac{9}{5}C+32$ to convert $35\mathrm{^{\circ}C}$ to degrees $F$