# 9.7 Higher roots  (Page 4/8)

 Page 4 / 8

Simplify: $\sqrt[5]{3x}+\sqrt[5]{3x}$ $3\sqrt[3]{9}-\sqrt[3]{9}$ .

$2\sqrt[5]{3x}$ $2\sqrt[3]{9}$

Simplify: $\sqrt[4]{10y}+\sqrt[4]{10y}$ $5\sqrt[6]{32}-3\sqrt[6]{32}$ .

$2\sqrt[4]{10y}$ $2\sqrt[6]{32}$

When an expression does not appear to have like radicals, we will simplify each radical first. Sometimes this leads to an expression with like radicals.

Simplify: $\sqrt[3]{54}-\sqrt[3]{16}$ $\sqrt[4]{48}+\sqrt[4]{243}$ .

## Solution

1. $\begin{array}{ccc}& & \sqrt[3]{54}-\sqrt[3]{16}\hfill \\ \\ \\ \text{Rewrite each radicand using perfect cube factors.}\hfill & & \sqrt[3]{27}·\sqrt[3]{2}-\sqrt[3]{8}·\sqrt[3]{2}\hfill \\ \\ \\ \text{Rewrite the perfect cubes.}\hfill & & \sqrt[3]{{\left(3\right)}^{3}}\phantom{\rule{0.2em}{0ex}}\sqrt[3]{2}-\sqrt[3]{{\left(2\right)}^{3}}\phantom{\rule{0.2em}{0ex}}\sqrt[3]{2}\hfill \\ \\ \\ \text{Simplify the radicals where possible.}\hfill & & 3\sqrt[3]{2}-2\sqrt[3]{2}\hfill \\ \\ \\ \text{Combine like radicals.}\hfill & & \sqrt[3]{2}\hfill \end{array}$

2. $\begin{array}{ccc}& & \phantom{\rule{2em}{0ex}}\sqrt[4]{48}+\sqrt[4]{243}\hfill \\ \\ \\ \text{Rewrite using perfect fourth power factors.}\hfill & & \phantom{\rule{2em}{0ex}}\sqrt[4]{16}·\sqrt[4]{3}+\sqrt[4]{81}·\sqrt[4]{3}\hfill \\ \\ \\ \text{Rewrite the perfect fourth powers.}\hfill & & \phantom{\rule{2em}{0ex}}\sqrt[4]{{\left(2\right)}^{4}}\phantom{\rule{0.2em}{0ex}}\sqrt[4]{3}+\sqrt[4]{{\left(3\right)}^{4}}\phantom{\rule{0.2em}{0ex}}\sqrt[4]{3}\hfill \\ \\ \\ \text{Simplify the radicals where possible.}\hfill & & \phantom{\rule{2em}{0ex}}2\sqrt[4]{3}+3\sqrt[4]{3}\hfill \\ \\ \\ \text{Combine like radicals.}\hfill & & \phantom{\rule{2em}{0ex}}5\sqrt[4]{3}\hfill \end{array}$

Simplify: $\sqrt[3]{192}-\sqrt[3]{81}$ $\sqrt[4]{32}+\sqrt[4]{512}$ .

$\sqrt[3]{3}$ $6\sqrt[4]{2}$

Simplify: $\sqrt[3]{108}-\sqrt[3]{250}$ $\sqrt[5]{64}+\sqrt[5]{486}$ .

$\text{−}\sqrt[3]{2}$ $5\sqrt[5]{2}$

Simplify: $\sqrt[3]{24{x}^{4}}-\sqrt[3]{-81{x}^{7}}$ $\sqrt[4]{162{y}^{9}}+\sqrt[4]{516{y}^{5}}$ .

## Solution

1. $\begin{array}{ccc}& & \phantom{\rule{4em}{0ex}}\sqrt[3]{24{x}^{4}}-\sqrt[3]{-81{x}^{7}}\hfill \\ \\ \\ \text{Rewrite each radicand using perfect cube factors.}\hfill & & \phantom{\rule{4em}{0ex}}\sqrt[3]{8{x}^{3}}·\sqrt[3]{3x}-\sqrt[3]{-27{x}^{6}}·\sqrt[3]{3x}\hfill \\ \\ \\ \text{Rewrite the perfect cubes.}\hfill & & \phantom{\rule{4em}{0ex}}\sqrt[3]{{\left(2x\right)}^{3}}\phantom{\rule{0.2em}{0ex}}\sqrt[3]{3x}-\sqrt[3]{{\left(-3{x}^{2}\right)}^{3}}\phantom{\rule{0.2em}{0ex}}\sqrt[3]{3x}\hfill \\ \\ \\ \text{Simplify the radicals where possible.}\hfill & & \phantom{\rule{4em}{0ex}}2x\sqrt[3]{3x}-\left(-3{x}^{2}\sqrt[3]{3x}\right)\hfill \end{array}$

2. $\begin{array}{ccc}& & \sqrt[4]{162{y}^{9}}+\sqrt[4]{516{y}^{5}}\hfill \\ \\ \\ \text{Rewrite each radicand using perfect fourth power factors.}\hfill & & \sqrt[4]{81{y}^{8}}·\sqrt[4]{2y}+\sqrt[4]{256{y}^{4}}·\sqrt[4]{2y}\hfill \\ \\ \\ \text{Rewrite the perfect fourth powers.}\hfill & & \sqrt[4]{{\left(3{y}^{2}\right)}^{4}}·\sqrt[4]{2y}+\sqrt[4]{{\left(4y\right)}^{4}}·\sqrt[4]{2y}\hfill \\ \\ \\ \text{Simplify the radicals where possible.}\hfill & & 3{y}^{2}\sqrt[4]{2y}+4|y|\sqrt[4]{2y}\hfill \end{array}$

Simplify: $\sqrt[3]{32{y}^{5}}-\sqrt[3]{-108{y}^{8}}$ $\sqrt[4]{243{r}^{11}}+\sqrt[4]{768{r}^{10}}$ .

$2y\sqrt[3]{4{y}^{2}}+3{y}^{2}\sqrt[3]{4{y}^{2}}$ $3{r}^{2}\sqrt[4]{3{r}^{3}}+4{r}^{2}\sqrt[4]{3{r}^{2}}$

Simplify: $\sqrt[3]{40{z}^{7}}-\sqrt[3]{-135{z}^{4}}$ $\sqrt[4]{80{s}^{13}}+\sqrt[4]{1280{s}^{6}}$ .

$2{z}^{2}\sqrt[3]{5z}+3z\sqrt[3]{5z}$ $2|{s}^{3}|\sqrt[4]{5s}+4|s|\sqrt[4]{5s}$

Access these online resources for additional instruction and practice with simplifying higher roots.

## Key concepts

• 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
$\sqrt[n]{ab}=\sqrt[n]{a}·\sqrt[n]{b}\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}\sqrt[n]{a}·\sqrt[n]{b}=\sqrt[n]{ab}$
• Quotient Property of n th Roots
$\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$
• To combine like radicals, simply add or subtract the coefficients while keeping the radical the same.

## Practice makes perfect

Simplify Expressions with Higher Roots

In the following exercises, simplify.

$\sqrt[3]{216}$
$\sqrt[4]{256}$
$\sqrt[5]{32}$

$\sqrt[3]{27}$
$\sqrt[4]{16}$
$\sqrt[5]{243}$

$3$ $2$ $3$

$\sqrt[3]{512}$
$\sqrt[4]{81}$
$\sqrt[5]{1}$

$\sqrt[3]{125}$
$\sqrt[4]{1296}$
$\sqrt[5]{1024}$

$5$ $6$ $4$

$\sqrt[3]{-8}$
$\sqrt[4]{-81}$
$\sqrt[5]{-32}$

$\sqrt[3]{-64}$
$\sqrt[4]{-16}$
$\sqrt[5]{-243}$

$-4$ $\text{not real}$ $-3$

$\sqrt[3]{-125}$
$\sqrt[4]{-1296}$
$\sqrt[5]{-1024}$

$\sqrt[3]{-512}$
$\sqrt[4]{-81}$
$\sqrt[5]{-1}$

$-8$ not a real number $-1$

$\sqrt[5]{{u}^{5}}$
$\sqrt[8]{{v}^{8}}$

$\sqrt[3]{{a}^{3}}$

$a$ $|b|$

$\sqrt[4]{{y}^{4}}$
$\sqrt[7]{{m}^{7}}$

$\sqrt[8]{{k}^{8}}$
$\sqrt[6]{{p}^{6}}$

$|k|$ $|p|$

$\sqrt[3]{{x}^{9}}$
$\sqrt[4]{{y}^{12}}$

$\sqrt[5]{{a}^{10}}$
$\sqrt[3]{{b}^{27}}$

${a}^{2}$ ${b}^{9}$

$\sqrt[4]{{m}^{8}}$
$\sqrt[5]{{n}^{20}}$

$\sqrt[6]{{r}^{12}}$
$\sqrt[3]{{s}^{30}}$

${r}^{2}$ ${s}^{10}$

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

$\sqrt[3]{-8{c}^{9}}$
$\sqrt[3]{125{d}^{15}}$

$-2{c}^{3}$ $5{d}^{5}$

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

$\sqrt[7]{128{r}^{14}}$
$\sqrt[4]{81{s}^{24}}$

$2{r}^{2}$ $3{s}^{6}$

Use the Product Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

$\sqrt[3]{{r}^{5}}$ $\sqrt[4]{{s}^{10}}$

$\sqrt[5]{{u}^{7}}$ $\sqrt[6]{{v}^{11}}$

$u\sqrt[5]{{u}^{2}}$ $v\sqrt[6]{{v}^{5}}$

$\sqrt[4]{{m}^{5}}$ $\sqrt[8]{{n}^{10}}$

$\sqrt[5]{{p}^{8}}$ $\sqrt[3]{{q}^{8}}$

$p\sqrt[5]{{p}^{3}}$ ${q}^{2}\sqrt[3]{{q}^{2}}$

$\sqrt[4]{32}$ $\sqrt[5]{64}$

$\sqrt[3]{625}$ $\sqrt[6]{128}$

$5\sqrt[3]{5}$ $2\sqrt[6]{2}$

$\sqrt[5]{64}$ $\sqrt[3]{256}$

$\sqrt[4]{3125}$ $\sqrt[3]{81}$

$5\sqrt[4]{5}$ $3\sqrt[3]{3}$

$\sqrt[3]{108{x}^{5}}$ $\sqrt[4]{48{y}^{6}}$

$\sqrt[5]{96{a}^{7}}$ $\sqrt[3]{375{b}^{4}}$

$2a\sqrt[5]{3{a}^{2}}$ $5b\sqrt[3]{3b}$

$\sqrt[4]{405{m}^{10}}$ $\sqrt[5]{160{n}^{8}}$

$\sqrt[3]{512{p}^{5}}$ $\sqrt[4]{324{q}^{7}}$

$8p\sqrt[3]{{p}^{2}}$ $3q\sqrt[4]{4{q}^{3}}$

$\sqrt[3]{-864}$ $\sqrt[4]{-256}$

$\sqrt[5]{-486}$ $\sqrt[6]{-64}$

$-3\sqrt[5]{2}$ $\text{not real}$

$\sqrt[5]{-32}$ $\sqrt[8]{-1}$

$\sqrt[3]{-8}$ $\sqrt[4]{-16}$

$-2$ $\text{not real}$

Use the Quotient Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

$\sqrt[3]{\frac{{p}^{11}}{{p}^{2}}}$ $\sqrt[4]{\frac{{q}^{17}}{{q}^{13}}}$

$\sqrt[5]{\frac{{d}^{12}}{{d}^{7}}}$ $\sqrt[8]{\frac{{m}^{12}}{{m}^{4}}}$

$d$ $|m|$

$\sqrt[5]{\frac{{u}^{21}}{{u}^{11}}}$ $\sqrt[6]{\frac{{v}^{30}}{{v}^{12}}}$

$\sqrt[3]{\frac{{r}^{14}}{{r}^{5}}}$ $\sqrt[4]{\frac{{c}^{21}}{{c}^{9}}}$

${r}^{2}$ $|{c}^{3}|$

$\frac{\sqrt[4]{64}}{\sqrt[4]{2}}$ $\frac{\sqrt[5]{128{x}^{8}}}{\sqrt[5]{2{x}^{2}}}$

$\frac{\sqrt[3]{-625}}{\sqrt[3]{5}}$ $\frac{\sqrt[4]{80{m}^{7}}}{\sqrt[4]{5m}}$

$-5$ $4m\sqrt[4]{{m}^{2}}$

$\sqrt[3]{\frac{1050}{2}}$ $\sqrt[4]{\frac{486{y}^{9}}{2{y}^{3}}}$

$\sqrt[3]{\frac{162}{6}}$ $\sqrt[4]{\frac{160{r}^{10}}{5{r}^{3}}}$

$3\sqrt[3]{6}$ $2|r|\sqrt[4]{2{r}^{3}}$

$\sqrt[3]{\frac{54{a}^{8}}{{b}^{3}}}$ $\sqrt[4]{\frac{64{c}^{5}}{{d}^{2}}}$

$\sqrt[5]{\frac{96{r}^{11}}{{s}^{3}}}$ $\sqrt[6]{\frac{128{u}^{7}}{{v}^{3}}}$

$\frac{2{r}^{2}\sqrt[5]{3r}}{{s}^{3}}$ $\frac{2{u}^{3}\sqrt[6]{2uv3}}{v}$

$\sqrt[3]{\frac{81{s}^{8}}{{t}^{3}}}$ $\sqrt[4]{\frac{64{p}^{15}}{{q}^{12}}}$

$\sqrt[3]{\frac{625{u}^{10}}{{v}^{3}}}$ $\sqrt[4]{\frac{729{c}^{21}}{{d}^{8}}}$

$\frac{5{u}^{3}\sqrt[3]{5u}}{v}$ $\frac{3{c}^{5}\sqrt[4]{9c}}{{d}^{2}}$

In the following exercises, simplify.

$\sqrt[7]{8p}+\sqrt[7]{8p}$
$3\sqrt[3]{25}-\sqrt[3]{25}$

$\sqrt[3]{15q}+\sqrt[3]{15q}$
$2\sqrt[4]{27}-6\sqrt[4]{27}$

$2\sqrt[3]{15q}$ $-4\sqrt[4]{27}$

$3\sqrt[5]{9x}+7\sqrt[5]{9x}$
$8\sqrt[7]{3q}-2\sqrt[7]{3q}$

$\sqrt[3]{81}-\sqrt[3]{192}$
$\sqrt[4]{512}-\sqrt[4]{32}$

$\sqrt[3]{250}-\sqrt[3]{54}$
$\sqrt[4]{243}-\sqrt[4]{1875}$

$5\sqrt[3]{5}-3\sqrt[3]{2}$ $-2\sqrt[4]{3}$

$\sqrt[3]{128}+\sqrt[3]{250}$
$\sqrt[5]{729}+\sqrt[5]{96}$

$\sqrt[4]{243}+\sqrt[4]{1250}$
$\sqrt[3]{2000}+\sqrt[3]{54}$

$3\sqrt[4]{3}+5\sqrt[4]{2}$ $13\sqrt[3]{2}$

$\sqrt[3]{64{a}^{10}}-\sqrt[3]{-216{a}^{12}}$
$\sqrt[4]{486{u}^{7}}+\sqrt[4]{768{u}^{3}}$

$\sqrt[3]{80{b}^{5}}-\sqrt[3]{-270{b}^{3}}$
$\sqrt[4]{160{v}^{10}}-\sqrt[4]{1280{v}^{3}}$

$2b\sqrt[3]{10{b}^{2}}+3b\sqrt[3]{10}$ $2{v}^{2}\sqrt[4]{10{v}^{2}}-4\sqrt[4]{5{v}^{3}}$

Mixed Practice

In the following exercises, simplify.

$\sqrt[4]{16}$

$\sqrt[6]{64}$

$2$

$\sqrt[3]{{a}^{3}}$

$|b|$

$\sqrt[3]{-8{c}^{9}}$

$\sqrt[3]{125{d}^{15}}$

$5{d}^{5}$

$\sqrt[3]{{r}^{5}}$

$\sqrt[4]{{s}^{10}}$

${s}^{2}\sqrt[4]{{s}^{2}}$

$\sqrt[3]{108{x}^{5}}$

$\sqrt[4]{48{y}^{6}}$

$2y\sqrt[4]{3{y}^{2}}$

$\sqrt[5]{-486}$

$\sqrt[6]{-64}$

$\text{not real}$

$\frac{\sqrt[4]{64}}{\sqrt[4]{2}}$

$\frac{\sqrt[5]{128{x}^{8}}}{\sqrt[5]{2{x}^{2}}}$

$2x\sqrt[5]{2x}$

$\sqrt[5]{\frac{96{r}^{11}}{{s}^{3}}}$

$\sqrt[6]{\frac{128{u}^{7}}{{v}^{3}}}$

$\frac{2{u}^{3}\sqrt[6]{2uv3}}{v}$

$\sqrt[3]{81}-\sqrt[3]{192}$

$\sqrt[4]{512}-\sqrt[4]{32}$

$4\sqrt[4]{2}$

$\sqrt[3]{64{a}^{10}}-\sqrt[3]{-216{a}^{12}}$

$\sqrt[4]{486{u}^{7}}+\sqrt[4]{768{u}^{3}}$

$3u\sqrt[4]{6{u}^{3}}+4\sqrt[4]{3{u}^{3}}$

## Everyday math

Population growth The expression $10·{x}^{n}$ models the growth of a mold population after $n$ generations. There were 10 spores at the start, and each had $x$ offspring. So $10·{x}^{n}$ is the number of offspring at the fifth generation. At the fifth generation there were 10,240 offspring. Simplify the expression $\sqrt[5]{\frac{10,240}{10}}$ to determine the number of offspring of each spore.

Spread of a virus The expression $3·{x}^{n}$ models the spread of a virus after $n$ cycles. There were three people originally infected with the virus, and each of them infected $x$ people. So $3·{x}^{4}$ is the number of people infected on the fourth cycle. At the fourth cycle 1875 people were infected. Simplify the expression $\sqrt[4]{\frac{1875}{3}}$ to determine the number of people each person infected.

$5$

## Writing exercises

Explain how you know that $\sqrt[5]{{x}^{10}}={x}^{2}$ .

Explain why $\sqrt[4]{-64}$ is not a real number but $\sqrt[3]{-64}$ is.

## Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

What does this checklist tell you about your mastery of this section? What steps will you take to improve?

Larry and Tom were standing next to each other in the backyard when Tom challenged Larry to guess how tall he was. Larry knew his own height is 6.5 feet and when they measured their shadows, Larry’s shadow was 8 feet and Tom’s was 7.75 feet long. What is Tom’s height?
6.25
Ciid
Wayne is hanging a string of lights 57 feet long around the three sides of his patio, which is adjacent to his house. the length of his patio, the side along the house, is 5 feet longer than twice it's width. Find the length and width of the patio.
Ciid
tyler
tyler
tyler
tyler
SOH = Sine is Opposite over Hypotenuse. CAH= Cosine is Adjacent over Hypotenuse. TOA = Tangent is Opposite over Adjacent.
tyler
H=57 and O=285 figure out what the adjacent?
tyler
Amara currently sells televisions for company A at a salary of $17,000 plus a$100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a$20 commission for each television she sells. How many televisions would Amara need to sell for the options to be equal?
what is the quantity and price of the televisions for both options?
karl
Amara has to sell 120 televisions to make 29,000 of the salary of company B. 120 * TV 100= commission 12,000+ 17,000 = of company a salary 29,000 17000+
Ciid
Amara has to sell 120 televisions to make 29,000 of the salary of company B. 120 * TV 100= commission 12,000+ 17,000 = of company a salary 29,000
Ciid
I'm mathematics teacher from highly recognized university.
here a question professor How many soldiers are there in a group of 27 sailors and soldiers if there are four fifths as many sailors as soldiers? can you write out the college you went to with the name of the school you teach at and let me know the answer I've got it to be honest with you
tyler
is anyone else having issues with the links not doing anything?
Yes
Val
chapter 1 foundations 1.2 exercises variables and algebraic symbols
June needs 45 gallons of punch for a party and has 2 different coolers to carry it in. The bigger cooler is 5 times as large as the smaller cooler. How many gallons can each cooler hold? Enter the answers in decimal form.
Joseph would like to make 12 pounds of a coffee blend at a cost of $6.25 per pound. He blends Ground Chicory at$4.40 a pound with Jamaican Blue Mountain at $8.84 per pound. How much of each type of coffee should he use? Samer 4x6.25=$25 coffee blend 4×4.40= $17.60 ground chicory 4x8.84= 35.36 blue mountain. In total they will spend for 12 pounds$77.96 they will spend in total
tyler
DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for three-quarters of an hour and Fabian rode his bike for half an hour to get to the park. Fabian’s speed was six miles per hour faster than DaMarcus’ speed. Find the speed of both soccer players.
i need help how to do this is confusing
what kind of math is it?
Danteii
help me to understand
huh, what is the algebra problem
Daniel
How many soldiers are there in a group of 27 sailors and soldiers if there are four fifths many sailors as soldiers?
tyler
What is the domain and range of heaviside
What is the domain and range of Heaviside and signum
Christopher
25-35
Fazal
The hypotenuse of a right triangle is 10cm long. One of the triangle’s legs is three times the length of the other leg. Find the lengths of the three sides of the triangle.
Tickets for a show are $70 for adults and$50 for children. For one evening performance, a total of 300 tickets were sold and the receipts totaled \$17,200. How many adult tickets and how many child tickets were sold?
A 50% antifreeze solution is to be mixed with a 90% antifreeze solution to get 200 liters of a 80% solution. How many liters of the 50% solution and how many liters of the 90% solution will be used?
June needs 45 gallons of punch for a party and has 2 different coolers to carry it in. The bigger cooler is 5 times as large as the smaller cooler. How many gallons can each cooler hold?