Chapter 8 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.

##

Worked example

complex conjugate pair

a

bi

a

bi

complex number

a

bi

a

b

a

b

complex number system

The complex number system is made up of both the real numbers and the imaginary numbers.

imaginary unit

i

2

like radicals

Like radicals are radical expressions with the same index and the same radicand.

radical equation

An equation in which a variable is in the radicand of a radical expression is called a radical equation.

radical function

A radical function is a function that is defined by a radical expression.

rationalizing the denominator

Rationalizing the denominator is the process of converting a fraction with a radical in the denominator to an equivalent fraction whose denominator is an integer.

square of a number

n

2

m

m

n

square root of a number

n

2

m,

n

m

standard form

a, b

##

Key Concepts

• Square Root Notation $\sqrt{m}$ is read ‘the square root of m’ If n${}^{2}$ = m, then $n=\sqrt{m},$ for $n\ge{}0.$ The square root of m, $\sqrt{m},$ is a positive number whose square is m.

• n${}^{th}$ Root of a Number If ${b}^{n}=a,$ then b is an $n^{th}$ root of a. The principal $n^{th}$ root of a is written $\sqrt[n]{a}.$ n is called the index of the radical.

• Properties of $\sqrt[n]{a}$ When n is an even number and $a\ge{}0,$ then $\sqrt[n]{a}$ is a real number $a<0,$ then $\sqrt[n]{a}$ is not a real number When n is an odd number, $\sqrt[n]{a}$ is a real number for all values of a.

• Simplifying Odd and Even Roots For any integer $n\ge{}2,$ when n is odd $\sqrt[n]{{a}^{n}}=a$ when n is even $\sqrt[n]{{a}^{n}}=|a|$ We must use the absolute value signs when we take an even root of an expression with a variable in the radical.

• Simplified Radical Expression For real numbers a, m and $n\ge{}2$$\sqrt[n]{a}$ is considered simplified if a has no factors of ${m}^{n}$

• Product Property of $n^{th}$ Roots For any real numbers, $\sqrt[n]{a}$ and $\sqrt[n]{b},$ and for any integer $n\ge{}2$$\sqrt[n]{ab}=\sqrt[n]{a}\cdot\sqrt[n]{b}$ and $\sqrt[n]{a}\cdot\sqrt[n]{b}=\sqrt[n]{ab}$

• How to simplify a radical expression using the Product Property Step 1. Find the largest factor in the radicand that is a perfect power of the index.Rewrite the radicand as a product of two factors, using that factor. Step 2. Use the product rule to rewrite the radical as the product of two radicals. Step 3. Simplify the root of the perfect power.

• Quotient Property of Radical Expressions If $\sqrt[n]{a}$ and $\sqrt[n]{b}$ are real numbers, $b\ne{}0,$ and for any integer $n\ge{}2$ then, $\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}$ and $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$

• How to simplify a radical expression using the Quotient Property. Step 1. Simplify the fraction in the radicand, if possible. Step 2. Use the Quotient Property to rewrite the radical as the quotient of two radicals. Step 3. Simplify the radicals in the numerator and the denominator.

• Rational Exponent ${a}^{\frac{1}{n}}$ If $\sqrt[n]{a}$ is a real number and $n\ge{}2,$ then ${a}^{\frac{1}{n}}=\sqrt[n]{a}.$

• Rational Exponent ${a}^{\frac{m}{n}}$ For any positive integers m and n, ${a}^{\frac{m}{n}}={(\sqrt[n]{a})}^{m}$ and ${a}^{\frac{m}{n}}=\sqrt[n]{{a}^{m}}$

• Properties of Exponents If a, b are real numbers and m, n are rational numbers, then Product Property ${a}^{m}\cdot{a}^{n}={a}^{m+n}$ Power Property ${({a}^{m})}^{n}={a}^{m\cdot{}n}$ Product to a Power ${(ab)}^{m}={a}^{m}{b}^{m}$ Quotient Property $\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},\,a\ne{}0$ Zero Exponent Definition ${a}^{0}=1,$$a\ne{}0$ Quotient to a Power Property ${(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}},\,b\ne{}0$ Negative Exponent Property ${a}^{-n}=\frac{1}{{a}^{n}},a\ne{}0$

• Product Property of Roots For any real numbers, $\sqrt[n]{a}$ and $\sqrt[n]{b},$ and for any integer $n\ge{}2$$\sqrt[n]{ab}=\sqrt[n]{a}\cdot\sqrt[n]{b}$ and $\sqrt[n]{a}\cdot\sqrt[n]{b}=\sqrt[n]{ab}$

• Special Products $\begin{array}{cccccc} \mathrm{Binomial Squares} & & & & & \mathrm{Product of Conjugates} \\ {(a+b)}^{2}={a}^{2}+2ab+{b}^{2} & & & & & (a+b)(a-b)={a}^{2}-{b}^{2} \\ {(a-b)}^{2}={a}^{2}-2ab+{b}^{2} & & & & & \\ \end{array}$

• Quotient Property of Radical Expressions If $\sqrt[n]{a}$ and $\sqrt[n]{b}$ are real numbers, $b\ne{}0,$ and for any integer $n\ge{}2$ then, $\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}$ and $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$

• Simplified Radical Expressions A radical expression is considered simplified if there are: no factors in the radicand that have perfect powers of the index no fractions in the radicand no radicals in the denominator of a fraction

• Binomial Squares $\begin{array}{c} {(a+b)}^{2}={a}^{2}+2ab+{b}^{2} \\ {(a-b)}^{2}={a}^{2}-2ab+{b}^{2} \\ \end{array}$

• Solve a Radical Equation Step 1. Isolate one of the radical terms on one side of the equation. Step 2. Raise both sides of the equation to the power of the index. Step 3. Are there any more radicals? If yes, repeat Step 1 and Step 2 again. If no, solve the new equation. Step 4. Check the answer in the original equation.

• Problem Solving Strategy for Applications with Formulas Step 1. Read the problem and make sure all the words and ideas are understood. When appropriate, draw a figure and label it with the given information. Step 2. Identify what we are looking for. Step 3. Name what we are looking for by choosing a variable to represent it. Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information. Step 5. Solve the equation using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

• Falling Objects On Earth, if an object is dropped from a height of h feet, the time in seconds it will take to reach the ground is found by using the formula $t=\frac{\sqrt{h}}{4}.$

• Skid Marks and Speed of a Car If the length of the skid marks is d feet, then the speed, s, of the car before the brakes were applied can be found by using the formula $s=\sqrt{24d}.$

• Properties of $\sqrt[n]{a}$ When n is an even number and: $a\ge{}0,$ then $\sqrt[n]{a}$ is a real number. $a<0,$ then $\sqrt[n]{a}$ is not a real number. When n is an odd number, $\sqrt[n]{a}$ is a real number for all values of a.

• Domain of a Radical Function When the index of the radical is even, the radicand must be greater than or equal to zero. When the index of the radical is odd, the radicand can be any real number.

• Square Root of a Negative Number If b is a positive real number, then $\sqrt{-b}=\sqrt{b}i$$a+bi$$b=0$$\begin{array}{c} a+0\cdot{}i \\ \,a \\ \end{array}$ Real number $b\ne{}0$$a+bi$ Imaginary number $a=0$$\begin{array}{c} 0+bi \\ \,bi \\ \end{array}$ Pure imaginary numberTable 8.1 A complex number is in standard form when written as a + bi, where a, b are real numbers.

• Product of Complex Conjugates If a, b are real numbers, then $(a-bi)(a+bi)={a}^{2}+{b}^{2}$

• How to Divide Complex Numbers Step 1. Write both the numerator and denominator in standard form. Step 2. Multiply the numerator and denominator by the complex conjugate of the denominator. Step 3. Simplify and write the result in standard form.

##

Review Exercises

Simplify Expressions with Roots

In the following exercises, simplify.

Estimate and Approximate Roots

In the following exercises, estimate each root between two consecutive whole numbers.

In the following exercises, approximate each root and round to two decimal places.

Simplify Variable Expressions with Roots

In the following exercises, simplify using absolute values as necessary.

Use the Product Property to Simplify Radical Expressions

In the following exercises, use the Product Property to simplify radical expressions.

In the following exercises, simplify using absolute value signs as needed.

Use the Quotient Property to Simplify Radical Expressions

In the following exercises, use the Quotient Property to simplify square roots.

Simplify expressions with ${a}^{\frac{1}{n}}$

In the following exercises, write as a radical expression.

In the following exercises, write with a rational exponent.

In the following exercises, simplify.

Simplify Expressions with ${a}^{\frac{m}{n}}$

In the following exercises, write with a rational exponent.

In the following exercises, simplify.

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

Add and Subtract Radical Expressions

In the following exercises, simplify.

Multiply Radical Expressions

In the following exercises, simplify.

Use Polynomial Multiplication to Multiply Radical Expressions

In the following exercises, multiply.

Divide Square Roots

In the following exercises, simplify.

Rationalize a One Term Denominator

In the following exercises, rationalize the denominator.

Rationalize a Two Term Denominator

In the following exercises, simplify.

Solve Radical Equations

In the following exercises, solve.

Solve Radical Equations with Two Radicals

In the following exercises, solve.

Use Radicals in Applications

In the following exercises, solve. Round approximations to one decimal place.

Evaluate a Radical Function

In the following exercises, evaluate each function.

Find the Domain of a Radical Function

In the following exercises, find the domain of the function and write the domain in interval notation.

Graph Radical Functions

In the following exercises, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

Evaluate the Square Root of a Negative Number

In the following exercises, write each expression in terms of i and simplify if possible.

Add or Subtract Complex Numbers

In the following exercises, add or subtract.

Multiply Complex Numbers

In the following exercises, multiply.

In the following exercises, multiply using the Product of Binomial Squares Pattern.

In the following exercises, multiply using the Product of Complex Conjugates Pattern.

Divide Complex Numbers

In the following exercises, divide.

Simplify Powers of i

In the following exercises, simplify.

##

In the following exercises, simplify using absolute values as necessary.

In the following exercises, simplify. Assume all variables are positive.

In the following exercises, solve.

In the following exercise, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

Key Terms

complex conjugate pair

a

bi

a

bi

complex number

a

bi

a

b

a

b

complex number system

The complex number system is made up of both the real numbers and the imaginary numbers.

imaginary unit

i

2

like radicals

Like radicals are radical expressions with the same index and the same radicand.

radical equation

An equation in which a variable is in the radicand of a radical expression is called a radical equation.

radical function

A radical function is a function that is defined by a radical expression.

rationalizing the denominator

Rationalizing the denominator is the process of converting a fraction with a radical in the denominator to an equivalent fraction whose denominator is an integer.

square of a number

n

2

m

m

n

square root of a number

n

2

m,

n

m

standard form

a, b

##

Key Concepts

• Square Root Notation $\sqrt{m}$ is read ‘the square root of m’ If n${}^{2}$ = m, then $n=\sqrt{m},$ for $n\ge{}0.$ The square root of m, $\sqrt{m},$ is a positive number whose square is m.

• n${}^{th}$ Root of a Number If ${b}^{n}=a,$ then b is an $n^{th}$ root of a. The principal $n^{th}$ root of a is written $\sqrt[n]{a}.$ n is called the index of the radical.

• Properties of $\sqrt[n]{a}$ When n is an even number and $a\ge{}0,$ then $\sqrt[n]{a}$ is a real number $a<0,$ then $\sqrt[n]{a}$ is not a real number When n is an odd number, $\sqrt[n]{a}$ is a real number for all values of a.

• Simplifying Odd and Even Roots For any integer $n\ge{}2,$ when n is odd $\sqrt[n]{{a}^{n}}=a$ when n is even $\sqrt[n]{{a}^{n}}=|a|$ We must use the absolute value signs when we take an even root of an expression with a variable in the radical.

• Simplified Radical Expression For real numbers a, m and $n\ge{}2$$\sqrt[n]{a}$ is considered simplified if a has no factors of ${m}^{n}$

• Product Property of $n^{th}$ Roots For any real numbers, $\sqrt[n]{a}$ and $\sqrt[n]{b},$ and for any integer $n\ge{}2$$\sqrt[n]{ab}=\sqrt[n]{a}\cdot\sqrt[n]{b}$ and $\sqrt[n]{a}\cdot\sqrt[n]{b}=\sqrt[n]{ab}$

• How to simplify a radical expression using the Product Property Step 1. Find the largest factor in the radicand that is a perfect power of the index.Rewrite the radicand as a product of two factors, using that factor. Step 2. Use the product rule to rewrite the radical as the product of two radicals. Step 3. Simplify the root of the perfect power.

• Quotient Property of Radical Expressions If $\sqrt[n]{a}$ and $\sqrt[n]{b}$ are real numbers, $b\ne{}0,$ and for any integer $n\ge{}2$ then, $\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}$ and $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$

• How to simplify a radical expression using the Quotient Property. Step 1. Simplify the fraction in the radicand, if possible. Step 2. Use the Quotient Property to rewrite the radical as the quotient of two radicals. Step 3. Simplify the radicals in the numerator and the denominator.

• Rational Exponent ${a}^{\frac{1}{n}}$ If $\sqrt[n]{a}$ is a real number and $n\ge{}2,$ then ${a}^{\frac{1}{n}}=\sqrt[n]{a}.$

• Rational Exponent ${a}^{\frac{m}{n}}$ For any positive integers m and n, ${a}^{\frac{m}{n}}={(\sqrt[n]{a})}^{m}$ and ${a}^{\frac{m}{n}}=\sqrt[n]{{a}^{m}}$

• Properties of Exponents If a, b are real numbers and m, n are rational numbers, then Product Property ${a}^{m}\cdot{a}^{n}={a}^{m+n}$ Power Property ${({a}^{m})}^{n}={a}^{m\cdot{}n}$ Product to a Power ${(ab)}^{m}={a}^{m}{b}^{m}$ Quotient Property $\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},\,a\ne{}0$ Zero Exponent Definition ${a}^{0}=1,$$a\ne{}0$ Quotient to a Power Property ${(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}},\,b\ne{}0$ Negative Exponent Property ${a}^{-n}=\frac{1}{{a}^{n}},a\ne{}0$

• Product Property of Roots For any real numbers, $\sqrt[n]{a}$ and $\sqrt[n]{b},$ and for any integer $n\ge{}2$$\sqrt[n]{ab}=\sqrt[n]{a}\cdot\sqrt[n]{b}$ and $\sqrt[n]{a}\cdot\sqrt[n]{b}=\sqrt[n]{ab}$

• Special Products $\begin{array}{cccccc} \mathrm{Binomial Squares} & & & & & \mathrm{Product of Conjugates} \\ {(a+b)}^{2}={a}^{2}+2ab+{b}^{2} & & & & & (a+b)(a-b)={a}^{2}-{b}^{2} \\ {(a-b)}^{2}={a}^{2}-2ab+{b}^{2} & & & & & \\ \end{array}$

• Quotient Property of Radical Expressions If $\sqrt[n]{a}$ and $\sqrt[n]{b}$ are real numbers, $b\ne{}0,$ and for any integer $n\ge{}2$ then, $\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}$ and $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$

• Simplified Radical Expressions A radical expression is considered simplified if there are: no factors in the radicand that have perfect powers of the index no fractions in the radicand no radicals in the denominator of a fraction

• Binomial Squares $\begin{array}{c} {(a+b)}^{2}={a}^{2}+2ab+{b}^{2} \\ {(a-b)}^{2}={a}^{2}-2ab+{b}^{2} \\ \end{array}$

• Solve a Radical Equation Step 1. Isolate one of the radical terms on one side of the equation. Step 2. Raise both sides of the equation to the power of the index. Step 3. Are there any more radicals? If yes, repeat Step 1 and Step 2 again. If no, solve the new equation. Step 4. Check the answer in the original equation.

• Problem Solving Strategy for Applications with Formulas Step 1. Read the problem and make sure all the words and ideas are understood. When appropriate, draw a figure and label it with the given information. Step 2. Identify what we are looking for. Step 3. Name what we are looking for by choosing a variable to represent it. Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information. Step 5. Solve the equation using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

• Falling Objects On Earth, if an object is dropped from a height of h feet, the time in seconds it will take to reach the ground is found by using the formula $t=\frac{\sqrt{h}}{4}.$

• Skid Marks and Speed of a Car If the length of the skid marks is d feet, then the speed, s, of the car before the brakes were applied can be found by using the formula $s=\sqrt{24d}.$

• Properties of $\sqrt[n]{a}$ When n is an even number and: $a\ge{}0,$ then $\sqrt[n]{a}$ is a real number. $a<0,$ then $\sqrt[n]{a}$ is not a real number. When n is an odd number, $\sqrt[n]{a}$ is a real number for all values of a.

• Domain of a Radical Function When the index of the radical is even, the radicand must be greater than or equal to zero. When the index of the radical is odd, the radicand can be any real number.

• Square Root of a Negative Number If b is a positive real number, then $\sqrt{-b}=\sqrt{b}i$$a+bi$$b=0$$\begin{array}{c} a+0\cdot{}i \\ \,a \\ \end{array}$ Real number $b\ne{}0$$a+bi$ Imaginary number $a=0$$\begin{array}{c} 0+bi \\ \,bi \\ \end{array}$ Pure imaginary numberTable 8.1 A complex number is in standard form when written as a + bi, where a, b are real numbers.

• Product of Complex Conjugates If a, b are real numbers, then $(a-bi)(a+bi)={a}^{2}+{b}^{2}$

• How to Divide Complex Numbers Step 1. Write both the numerator and denominator in standard form. Step 2. Multiply the numerator and denominator by the complex conjugate of the denominator. Step 3. Simplify and write the result in standard form.

##

Review Exercises

Simplify Expressions with Roots

In the following exercises, simplify.

Estimate and Approximate Roots

In the following exercises, estimate each root between two consecutive whole numbers.

In the following exercises, approximate each root and round to two decimal places.

Simplify Variable Expressions with Roots

In the following exercises, simplify using absolute values as necessary.

Use the Product Property to Simplify Radical Expressions

In the following exercises, use the Product Property to simplify radical expressions.

In the following exercises, simplify using absolute value signs as needed.

Use the Quotient Property to Simplify Radical Expressions

In the following exercises, use the Quotient Property to simplify square roots.

Simplify expressions with ${a}^{\frac{1}{n}}$

In the following exercises, write as a radical expression.

In the following exercises, write with a rational exponent.

In the following exercises, simplify.

Simplify Expressions with ${a}^{\frac{m}{n}}$

In the following exercises, write with a rational exponent.

In the following exercises, simplify.

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

Add and Subtract Radical Expressions

In the following exercises, simplify.

Multiply Radical Expressions

In the following exercises, simplify.

Use Polynomial Multiplication to Multiply Radical Expressions

In the following exercises, multiply.

Divide Square Roots

In the following exercises, simplify.

Rationalize a One Term Denominator

In the following exercises, rationalize the denominator.

Rationalize a Two Term Denominator

In the following exercises, simplify.

Solve Radical Equations

In the following exercises, solve.

Solve Radical Equations with Two Radicals

In the following exercises, solve.

Use Radicals in Applications

In the following exercises, solve. Round approximations to one decimal place.

Evaluate a Radical Function

In the following exercises, evaluate each function.

Find the Domain of a Radical Function

In the following exercises, find the domain of the function and write the domain in interval notation.

Graph Radical Functions

In the following exercises, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

Evaluate the Square Root of a Negative Number

In the following exercises, write each expression in terms of i and simplify if possible.

Add or Subtract Complex Numbers

In the following exercises, add or subtract.

Multiply Complex Numbers

In the following exercises, multiply.

In the following exercises, multiply using the Product of Binomial Squares Pattern.

In the following exercises, multiply using the Product of Complex Conjugates Pattern.

Divide Complex Numbers

In the following exercises, divide.

Simplify Powers of i

In the following exercises, simplify.

##

In the following exercises, simplify using absolute values as necessary.

In the following exercises, simplify. Assume all variables are positive.

In the following exercises, solve.

In the following exercise, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

1. ⓐ $\sqrt{225}$ ⓑ $-\sqrt{16}$

2. ⓐ $\sqrt[3]{8}$ ⓑ $\sqrt[4]{81}$ ⓒ $\sqrt[5]{243}$

3. ⓐ $\sqrt{68}$ ⓑ $\sqrt[3]{84}$

4. ⓐ $\sqrt[3]{{a}^{3}}$ⓑ $\sqrt[7]{{b}^{7}}$

1. ⓐ $\sqrt[4]{{m}^{8}}$ⓑ $\sqrt[5]{{n}^{20}}$

2. ⓐ $\sqrt[3]{216{a}^{6}}$ⓑ $\sqrt[5]{32{b}^{20}}$

3. $\sqrt{125}$

4. ⓐ $\sqrt[3]{625}$ ⓑ $\sqrt[6]{128}$

1. ⓐ $\sqrt{80{s}^{15}}$ⓑ $\sqrt[5]{96{a}^{7}}$ⓒ $\sqrt[6]{128{b}^{7}}$

2. ⓐ $\sqrt[5]{-32}$ⓑ $\sqrt[8]{-1}$

3. ⓐ $\sqrt{\frac{72}{98}}$ ⓑ $\sqrt[3]{\frac{24}{81}}$ ⓒ $\sqrt[4]{\frac{6}{96}}$

4. $\sqrt{\frac{300{m}^{5}}{64}}$

1. ⓐ $\sqrt{\frac{27{p}^{2}q}{108{p}^{4}{q}^{3}}}$ⓑ $\sqrt[3]{\frac{16{c}^{5}{d}^{7}}{250{c}^{2}{d}^{2}}}$ⓒ $\sqrt[6]{\frac{2{m}^{9}{n}^{7}}{128{m}^{3}n}}$

2. ⓐ ${r}^{\frac{1}{2}}$ ⓑ ${s}^{\frac{1}{3}}$ ⓒ ${t}^{\frac{1}{4}}$

3. ⓐ ${625}^{\frac{1}{4}}$ⓑ ${243}^{\frac{1}{5}}$ⓒ ${32}^{\frac{1}{5}}$

4. ⓐ ${(-32)}^{\frac{1}{5}}$ⓑ ${(243)}^{-\frac{1}{5}}$ⓒ $-{125}^{\frac{1}{3}}$

1. ⓐ ${25}^{\frac{3}{2}}$ⓑ ${9}^{-\frac{3}{2}}$ⓒ ${(-64)}^{\frac{2}{3}}$

2. ⓐ ${6}^{\frac{5}{2}}\cdot{6}^{\frac{1}{2}}$ⓑ ${({b}^{15})}^{\frac{3}{5}}$ⓒ $\frac{{w}^{\frac{2}{7}}}{{w}^{\frac{9}{7}}}$

3. ⓐ $7\sqrt{2}-3\sqrt{2}$ⓑ $7\sqrt[3]{p}+2\sqrt[3]{p}$ⓒ $5\sqrt[3]{x}-3\sqrt[3]{x}$

4. ⓐ $\sqrt{48}+\sqrt{27}$ⓑ $\sqrt[3]{54}+\sqrt[3]{128}$ⓒ $6\sqrt[4]{5}-\frac{3}{2}\sqrt[4]{80}$

1. $3\sqrt{75{y}^{2}}+8y\sqrt{48}-\sqrt{300{y}^{2}}$

2. ⓐ $(3\sqrt{2{x}^{3}})(7\sqrt{18{x}^{2}})$ⓑ $(-6\sqrt[3]{20{a}^{2}})(-2\sqrt[3]{16{a}^{3}})$

3. ⓐ $(3-2\sqrt{7})(5-4\sqrt{7})$ⓑ $(\sqrt[3]{x}-5)(\sqrt[3]{x}-3)$

4. ⓐ ${(4+\sqrt{11})}^{2}$ⓑ ${(3-2\sqrt{5})}^{2}$

1. $(\sqrt[3]{3x}+2)(\sqrt[3]{3x}-2)$

2. ⓐ $\frac{\sqrt{320m{n}^{-5}}}{\sqrt{45{m}^{-7}{n}^{3}}}$ⓑ $\frac{\sqrt[3]{16{x}^{4}{y}^{-2}}}{\sqrt[3]{-54{x}^{-2}{y}^{4}}}$

3. ⓐ $\frac{1}{\sqrt[3]{11}}$ ⓑ $\sqrt[3]{\frac{7}{54}}$ ⓒ $\frac{3}{\sqrt[3]{3{x}^{2}}}$

4. $\frac{7}{2-\sqrt{6}}$

1. $\frac{\sqrt{x}+\sqrt{8}}{\sqrt{x}-\sqrt{8}}$

2. $\sqrt{5x+1}=-3$

3. $\sqrt{u-3}+3=u$

4. ${(8x+5)}^{\frac{1}{3}}+2=-1$

1. $2\sqrt{8r+1}-8=2$

2. $\sqrt[3]{2{x}^{2}+9x-18}=\sqrt[3]{{x}^{2}+3x-2}$

3. $\sqrt{x+1}-\sqrt{x-2}=1$

4. Accident investigation An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was 175 feet. Use the formula $s=\sqrt{24d}$ to find the speed of the vehicle before the brakes were applied. Round your answer to the nearest tenth.

1. $G(x)=\sqrt{5x-1},$ findⓐ $G(5)$ⓑ $G(2)$

2. For the function$g(x)=\sqrt[4]{4-4x},$ findⓐ $g(1)$ⓑ $g(-3)$

3. $F(x)=\sqrt{\frac{x+3}{x-2}}$

4. $F(x)=\sqrt[4]{10-7x}$

1. $g(x)=2\sqrt{x}$

2. $f(x)=\sqrt[3]{x}+3$

3. $\sqrt{-50}+\sqrt{-18}$

4. $(6+i)-(-2-4i)$

1. $(-2-5i)(-4+3i)$

2. $\sqrt{-4}\cdot\sqrt{-16}$

3. ${(-2-3i)}^{2}$

4. $\frac{2+i}{3-4i}$

1. $\sqrt[3]{125{x}^{9}}$

2. $\sqrt[3]{72{x}^{8}{y}^{4}}$

3. ⓐ ${256}^{-\frac{1}{4}}$ ⓑ $-{49}^{\frac{3}{2}}$

4. $\frac{{x}^{-\frac{1}{4}}\cdot{x}^{\frac{5}{4}}}{{x}^{-\frac{3}{4}}}$

1. $\sqrt{48{x}^{5}}-\sqrt{75{x}^{5}}$

2. $2\sqrt{12{x}^{5}}\cdot{}3\sqrt{6{x}^{3}}$

3. $(4-3\sqrt{3})(5+2\sqrt{3})$

4. $\frac{\sqrt{245x{y}^{-4}}}{\sqrt{45{x}^{-4}{y}^{3}}}$

1. $\frac{3}{2+\sqrt{3}}$

2. $-4i(-2-3i)$

3. ${i}^{172}$

4. $\sqrt{x+5}+1=x$

1. $g(x)=\sqrt{x+2}$