### Review Exercises

## Rational and Irrational Numbers

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

$6$

$2.9$

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

$0.42,0.\stackrel{\text{\u2013}}{\text{3}},\mathrm{2.56813\dots}$

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

ⓐ$\phantom{\rule{0.2em}{0ex}}\sqrt{49}$ ⓑ$\phantom{\rule{0.2em}{0ex}}\sqrt{55}$

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

$\mathrm{-9},0,\text{0.361....},\frac{8}{9},\sqrt{16},9$

$\mathrm{-5},\phantom{\rule{0.2em}{0ex}}-2\frac{1}{4},\phantom{\rule{0.2em}{0ex}}-\sqrt{4},\phantom{\rule{0.2em}{0ex}}0.\stackrel{\text{\u2014}}{25},\phantom{\rule{0.2em}{0ex}}\frac{13}{5},\phantom{\rule{0.2em}{0ex}}4$

## Commutative and Associative Properties

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

$6+4=\_\_\_\_$

$3n=\_\_\_\_$

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

$(13\xb75)\xb72=\_\_\_\_\_$

$(4+9x)+x=\_\_\_\_\_$

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

If $y=\frac{11}{12},$ evaluate:

ⓐ $\phantom{\rule{0.2em}{0ex}}y+0.7+(-y)$

ⓑ $\phantom{\rule{0.2em}{0ex}}y+(-y)+0.7$

If $z=-\frac{5}{3},$ evaluate:

ⓐ $\phantom{\rule{0.2em}{0ex}}z+5.39+(-z)$

ⓑ $\phantom{\rule{0.2em}{0ex}}z+(-z)+5.39$

If $k=65,$ evaluate:

ⓐ $\phantom{\rule{0.2em}{0ex}}\frac{4}{9}\left(\frac{9}{4}k\right)$

ⓑ $\phantom{\rule{0.2em}{0ex}}(\frac{4}{9}\xb7\frac{9}{4})k$

If $m=\mathrm{-13},$ evaluate:

ⓐ $\phantom{\rule{0.2em}{0ex}}-\frac{2}{5}\left(\frac{5}{2}m\right)$

ⓑ $\phantom{\rule{0.2em}{0ex}}(-\frac{2}{5}\xb7\frac{5}{2})m$

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

$6y+37+(\mathrm{-6}y)$

$\frac{14}{11}\xb7\frac{35}{9}\xb7\frac{14}{11}$

$\left(\frac{7}{12}+\frac{4}{5}\right)+\frac{1}{5}$

$\mathrm{-12}\left(4m\right)$

$11x+8y+16x+15y$

## Distributive Property

In the following exercises, simplify using the distributive property.

$7(x+9)$

$\mathrm{-3}(6m-1)$

$\frac{1}{3}(15n-6)$

$(a-4)-(6a+9)$

In the following exercises, evaluate using the distributive property.

If $u=2,$ evaluate

ⓐ $\phantom{\rule{0.2em}{0ex}}3(8u+9)\phantom{\rule{0.2em}{0ex}}\text{and}$

ⓑ $\phantom{\rule{0.2em}{0ex}}3\xb78u+3\xb79$ to show that $3(8u+9)=3\xb78u+3\xb79$

If $n=\frac{7}{8},$ evaluate

ⓐ$\phantom{\rule{0.2em}{0ex}}8(n+\frac{1}{4})$ and

ⓑ$\phantom{\rule{0.2em}{0ex}}8\xb7n+8\xb7\frac{1}{4}$ to show that $8(n+\frac{1}{4})=8\xb7n+8\xb7\frac{1}{4}$

If $d=14,$ evaluate

ⓐ $\mathrm{-100}\left(0.1d+0.35\right)$ and

ⓑ$\phantom{\rule{0.2em}{0ex}}\mathrm{-100}\xb7(0.1d)+\left(\mathrm{-100}\right)\left(0.35\right)$ to show that $\mathrm{-100}\left(0.1d+0.35\right)=\mathrm{-100}\xb7(0.1d)+\left(\mathrm{-100}\right)\left(0.35\right)$

If $y=\mathrm{-18},$ evaluate

ⓐ$\phantom{\rule{0.2em}{0ex}}-\left(y-18\right)$ and

ⓑ$\phantom{\rule{0.2em}{0ex}}-y+18$ to show that $-\left(y-18\right)=-y+18$

## Properties of Identities, Inverses, and Zero

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

$\mathrm{-35}(1)=\mathrm{-35}$

$(6x+0)+4x=6x+4x$

In the following exercises, find the additive inverse.

$\mathrm{-32}$

$\frac{3}{5}$

In the following exercises, find the multiplicative inverse.

$\frac{9}{2}$

$\frac{1}{10}$

In the following exercises, simplify.

$83\xb70$

$\frac{5}{0}$

$43+39+(\mathrm{-43})$

$\frac{5}{13}\xb757\xb7\frac{13}{5}$

$\frac{2}{3}\xb728\xb7\frac{3}{7}$

## Systems of Measurement

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

A floral arbor is $7$ feet tall. Convert the height to inches.

Kelly is $5$ feet $4$ inches tall. Convert her height to inches.

The height of Mount Shasta is $\mathrm{14,179}$ feet. Convert the height to miles.

The play lasted $1\frac{3}{4}$ hours. Convert the time to minutes.

Naomi’s baby weighed $5$ pounds $14$ ounces at birth. Convert the weight to ounces.

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

John caught $4$ lobsters. The weights of the lobsters were $1$ pound $9$ ounces, $1$ pound $12$ ounces, $4$ pounds $2$ ounces, and $2$ pounds $15$ ounces. What was the total weight of the lobsters?

Every day last week, Pedro recorded the amount of time he spent reading. He read for $50,25,83,45,32,60,\text{and}\phantom{\rule{0.2em}{0ex}}135$ minutes. How much time, in hours and minutes, did Pedro spend reading?

Fouad is $6$ feet $2$ inches tall. If he stands on a rung of a ladder $8$ feet $10$ inches high, how high off the ground is the top of Fouad’s head?

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?

In the following exercises, convert between metric units.

Donna is $1.7$ meters tall. Convert her height to centimeters.

One cup of yogurt contains $488$ milligrams of calcium. Convert this to grams.

Sergio weighed $2.9$ kilograms at birth. Convert this to grams.

In the following exercises, solve.

Minh is $2$ meters tall. His daughter is $88$ centimeters tall. How much taller, in meters, is Minh than his daughter?

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

One serving of cranberry juice contains $30$ grams of sugar. How many kilograms of sugar are in $30$ servings of cranberry juice?

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

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

Majid is $69$ inches tall. Convert his height to centimeters.

Caroline walked $2.5$ kilometers. Convert this length to miles.

Steve’s car holds $55$ liters of gas. Convert this to gallons.

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

$95\text{\xb0F}$

$20\text{\xb0F}$

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

$30\text{\xb0C}$

$\mathrm{-12}\text{\xb0C}$