Chapter 9 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

index

n

index

like radicals

Radicals with the same index and same radicand are called like radicals.

like square roots

Square roots with the same radicand are called like square roots.

n

n

n

n

radical equation

An equation in which the variable is in the radicand of a square root is called a radical equation

rational exponents

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

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

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 rationalizing the denominator.

square of a number

• If ${n}^{2}=m$, then $m$ is the square of $n$

square root notation

• If $m={n}^{2}$, then $\sqrt{m}=n$. We read $\sqrt{m}$ as ‘the square root of $m$.’

square root of a number

• If ${n}^{2}=m$, then $n$ is a square root of $m$

##

Key Concepts

• Note that the square root of a negative number is not a real number.

• Every positive number has two square roots, one positive and one negative. The positive square root of a positive number is the principal square root.

• We can estimate square roots using nearby perfect squares.

• We can approximate square roots using a calculator.

• When we use the radical sign to take the square root of a variable expression, we should specify that $x\ge{}0$ to make sure we get the principal square root.

• Simplified Square Root $\sqrt{a}$ is considered simplified if $a$ has no perfect-square factors.

• Product Property of Square Roots If a, b are non-negative real numbers, then

• Simplify a Square Root Using the Product Property To simplify a square root using the Product Property: Step 1. Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect square factor. Step 2. Use the product rule to rewrite the radical as the product of two radicals. Step 3. Simplify the square root of the perfect square.

• Quotient Property of Square Roots If a, b are non-negative real numbers and $b\ne{}0$, then

• Simplify a Square Root Using the Quotient Property To simplify a square root using the Quotient Property: Step 1. Simplify the fraction in the radicand, if possible. Step 2. Use the Quotient Rule to rewrite the radical as the quotient of two radicals. Step 3. Simplify the radicals in the numerator and the denominator.

• To add or subtract like square roots, add or subtract the coefficients and keep the like square root.

• Sometimes when we have to add or subtract square roots that do not appear to have like radicals, we find like radicals after simplifying the square roots.

• Product Property of Square Roots If a, b are nonnegative real numbers, then

• Special formulas for multiplying binomials and conjugates:

• The FOIL method can be used to multiply binomials containing radicals.

• Quotient Property of Square Roots If a, b are non-negative real numbers and $b\ne{}0$, then

• Simplified Square RootsA 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 Solve a Radical Equation: Step 1. Isolate the radical on one side of the equation. Step 2. Square both sides of the equation. Step 3. Solve the new equation. Step 4. Check the answer. Some solutions obtained may not work in the original equation.

• Solving 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.

• Area of a Square

• 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. For any integer $n\ge{}2$, when n is odd $\sqrt[n]{{a}^{n}}=a$ For any integer $n\ge{}2$, when n is even $\sqrt[n]{{a}^{n}}=|a|$

• $\sqrt[n]{a}$ is considered simplified if a has no factors of ${m}^{n}$.

• Product Property of *n*th Roots

• Quotient Property of *n*th Roots

• To combine like radicals, simply add or subtract the coefficients while keeping the radical the same.

• Summary of Exponent Properties

• 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:

Zero Exponent Definition ${a}^{0}=1$, $a\ne{}0$ Quotient to a Power Property ${(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}},\,b\ne{}0$

##

Review Exercises

Simplify Expressions with Square Roots

In the following exercises, simplify.

Estimate Square Roots

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

Approximate Square Roots

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

Simplify Variable Expressions with Square Roots

In the following exercises, simplify.

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

Add and Subtract Like Square Roots

In the following exercises, simplify.

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

Multiply Square Roots

In the following exercises, simplify.

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

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, rationalize the denominator.

Solve Radical Equations

In the following exercises, solve the equation.

Use Square Roots in Applications

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

Simplify Expressions with Higher Roots

In the following exercises, simplify.

Use the Product Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

Use the Quotient Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

Add and Subtract Higher Roots

In the following exercises, simplify.

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.

##

In the following exercises, simplify.

In the following exercises, rationalize the denominator.

In the following exercises, solve.

In the following exercise, solve.

Key Terms

index

n

index

like radicals

Radicals with the same index and same radicand are called like radicals.

like square roots

Square roots with the same radicand are called like square roots.

n

n

n

n

radical equation

An equation in which the variable is in the radicand of a square root is called a radical equation

rational exponents

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

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

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 rationalizing the denominator.

square of a number

• If ${n}^{2}=m$, then $m$ is the square of $n$

square root notation

• If $m={n}^{2}$, then $\sqrt{m}=n$. We read $\sqrt{m}$ as ‘the square root of $m$.’

square root of a number

• If ${n}^{2}=m$, then $n$ is a square root of $m$

##

Key Concepts

• Note that the square root of a negative number is not a real number.

• Every positive number has two square roots, one positive and one negative. The positive square root of a positive number is the principal square root.

• We can estimate square roots using nearby perfect squares.

• We can approximate square roots using a calculator.

• When we use the radical sign to take the square root of a variable expression, we should specify that $x\ge{}0$ to make sure we get the principal square root.

• Simplified Square Root $\sqrt{a}$ is considered simplified if $a$ has no perfect-square factors.

• Product Property of Square Roots If a, b are non-negative real numbers, then

• Simplify a Square Root Using the Product Property To simplify a square root using the Product Property: Step 1. Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect square factor. Step 2. Use the product rule to rewrite the radical as the product of two radicals. Step 3. Simplify the square root of the perfect square.

• Quotient Property of Square Roots If a, b are non-negative real numbers and $b\ne{}0$, then

• Simplify a Square Root Using the Quotient Property To simplify a square root using the Quotient Property: Step 1. Simplify the fraction in the radicand, if possible. Step 2. Use the Quotient Rule to rewrite the radical as the quotient of two radicals. Step 3. Simplify the radicals in the numerator and the denominator.

• To add or subtract like square roots, add or subtract the coefficients and keep the like square root.

• Sometimes when we have to add or subtract square roots that do not appear to have like radicals, we find like radicals after simplifying the square roots.

• Product Property of Square Roots If a, b are nonnegative real numbers, then

• Special formulas for multiplying binomials and conjugates:

• The FOIL method can be used to multiply binomials containing radicals.

• Quotient Property of Square Roots If a, b are non-negative real numbers and $b\ne{}0$, then

• Simplified Square RootsA 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 Solve a Radical Equation: Step 1. Isolate the radical on one side of the equation. Step 2. Square both sides of the equation. Step 3. Solve the new equation. Step 4. Check the answer. Some solutions obtained may not work in the original equation.

• Solving 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.

• Area of a Square

• 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. For any integer $n\ge{}2$, when n is odd $\sqrt[n]{{a}^{n}}=a$ For any integer $n\ge{}2$, when n is even $\sqrt[n]{{a}^{n}}=|a|$

• $\sqrt[n]{a}$ is considered simplified if a has no factors of ${m}^{n}$.

• Product Property of *n*th Roots

• Quotient Property of *n*th Roots

• To combine like radicals, simply add or subtract the coefficients while keeping the radical the same.

• Summary of Exponent Properties

• 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:

Zero Exponent Definition ${a}^{0}=1$, $a\ne{}0$ Quotient to a Power Property ${(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}},\,b\ne{}0$

##

Review Exercises

Simplify Expressions with Square Roots

In the following exercises, simplify.

Estimate Square Roots

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

Approximate Square Roots

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

Simplify Variable Expressions with Square Roots

In the following exercises, simplify.

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

Add and Subtract Like Square Roots

In the following exercises, simplify.

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

Multiply Square Roots

In the following exercises, simplify.

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

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, rationalize the denominator.

Solve Radical Equations

In the following exercises, solve the equation.

Use Square Roots in Applications

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

Simplify Expressions with Higher Roots

In the following exercises, simplify.

Use the Product Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

Use the Quotient Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

Add and Subtract Higher Roots

In the following exercises, simplify.

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.

##

In the following exercises, simplify.

In the following exercises, rationalize the denominator.

In the following exercises, solve.

In the following exercise, solve.

1. $-\sqrt{81}$

2. $\sqrt{-36}$

3. $\sqrt{155}$

4. $\sqrt{64{b}^{2}}$

5. $\sqrt{225{m}^{2}{n}^{2}}$

6. $\sqrt{49{y}^{2}}$

1. $\sqrt{121{c}^{2}{d}^{2}}$

2. $\sqrt{98}$

3. $\sqrt{{y}^{19}}$

4. $\sqrt{36{n}^{13}}$

5. $\sqrt{150{n}^{7}}$

6. $\sqrt{108{r}^{5}{s}^{3}}$

1. $\frac{6+\sqrt{72}}{6}$

2. $\sqrt{\frac{81}{36}}$

3. $\sqrt{\frac{{y}^{6}}{{y}^{2}}}$

4. $\sqrt{\frac{72{q}^{8}}{2{q}^{4}}}$

5. $\sqrt{\frac{26}{169}}$

6. $\sqrt{\frac{36{r}^{10}}{16{r}^{5}}}$

1. $\sqrt{\frac{12{r}^{5}{s}^{7}}{75{r}^{2}s}}$

2. $5\sqrt{5}+7\sqrt{5}$

3. $6\sqrt{m}-2\sqrt{m}$

4. $8\sqrt{13}+2\sqrt{3}+3\sqrt{13}$

5. $2\sqrt{3rs}+\sqrt{3rs}-5\sqrt{rs}$

6. $\sqrt{8}+3\sqrt{2}$

1. $\sqrt{48}+\sqrt{75}$

2. $\frac{1}{3}\sqrt{27}-\frac{1}{8}\sqrt{192}$

3. $6\sqrt{18{n}^{4}}-3\sqrt{8{n}^{4}}+{n}^{2}\sqrt{50}$

4. $2\sqrt{2}\cdot{}6\sqrt{14}$

5. $(6\sqrt{2y})(3\sqrt{50{y}^{3}})$

6. $10(2-\sqrt{7})$

1. $(5+\sqrt{2})(3-\sqrt{2})$

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

3. ${(1+6\sqrt{p})}^{2}$

4. $(3+2\sqrt{7})(3-2\sqrt{7})$

5. $\frac{\sqrt{75}}{10}$

6. $\frac{\sqrt{48}}{\sqrt{27}}$

1. $\frac{\sqrt{20{y}^{5}}}{\sqrt{2y}}$

2. $\frac{10}{\sqrt{15}}$

3. $\frac{5}{3\sqrt{5}}$

4. $\sqrt{\frac{3}{28}}$

5. $\frac{4}{4+\sqrt{27}}$

6. $\frac{4}{2-\sqrt{5}}$

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

2. $\sqrt{6m+4}-5=0$

3. $\sqrt{u-4}+4=u$

4. $\sqrt{v-9}+9=0$

5. $2\sqrt{2x-7}-4=8$

6. $\sqrt{u}+2=\sqrt{u+5}$

1. A helicopter dropped a package from a height of 900 feet above a stranded hiker. Use the formula $t=\frac{\sqrt{h}}{4}$ to find how many seconds it took for the package to reach the hiker.

2. ⓐ $\sqrt[6]{64}$ⓑ $\sqrt[3]{64}$

3. ⓐ $\sqrt[9]{{d}^{9}}$ⓑ $\sqrt[8]{{v}^{8}}$

4. ⓐ $\sqrt[4]{16{x}^{8}}$ⓑ $\sqrt[6]{64{y}^{12}}$

5. ⓐ $\sqrt[9]{{d}^{9}}$ⓑ

6. ⓐ $\sqrt[5]{64{c}^{8}}$ⓑ $\sqrt[4]{48{d}^{7}}$

1. ⓐ $\sqrt[3]{-500}$ⓑ $\sqrt[4]{-16}$

2. $\sqrt[3]{\frac{{w}^{12}}{{w}^{2}}}$

3. $\sqrt[3]{\frac{54{z}^{9}}{2{z}^{3}}}$

4. $4\sqrt[5]{20}-2\sqrt[5]{20}$

5. $\sqrt[4]{1250}-\sqrt[4]{162}$

6. $\sqrt[5]{96{t}^{8}}+\sqrt[5]{486{t}^{4}}$

1. ${s}^{\frac{1}{10}}$

2. $\sqrt[6]{v}$

3. $\sqrt[6]{10z}$

4. ${(125)}^{-\frac{1}{3}}$

5. ${(36)}^{-\frac{1}{2}}$

6. $\sqrt[5]{{n}^{8}}$

1. ${64}^{\frac{5}{2}}$

2. ${81}^{-\frac{5}{2}}$

3. ${({x}^{6})}^{\frac{4}{3}}$

4. ${(16{s}^{\frac{9}{4}})}^{\frac{1}{4}}$

5. $\frac{{z}^{\frac{2}{3}}\cdot{z}^{-\frac{1}{3}}}{{z}^{-\frac{5}{3}}}$

6. $\sqrt{169{m}^{4}{n}^{2}}$

1. $3\sqrt{13}+5\sqrt{2}+\sqrt{13}$

2. $(3\sqrt{6y})(2\sqrt{50{y}^{3}})$

3. ${(1-2\sqrt{q})}^{2}$

4. ⓐ $\sqrt[4]{81{x}^{12}}$ⓑ $\sqrt[6]{64{y}^{18}}$

5. $\sqrt{\frac{14{y}^{3}}{7y}}$

6. $\sqrt[4]{512}-2\sqrt[4]{32}$

1. $\frac{{w}^{\frac{3}{4}}}{{w}^{\frac{7}{4}}}$

2. $\frac{3}{2\sqrt{6}}$

3. A helicopter flying at an altitude of 600 feet dropped a package to a lifeboat. Use the formula $t=\frac{\sqrt{h}}{4}$ to find how many seconds it took for the package to reach the hiker. Round your answer to the nearest tenth of a second.