# 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?

will every polynomial have finite number of multiples?
a=# of 10's. b=# of 20's; a+b=54; 10a + 20b=$910; a=54 -b; 10(54-b) + 20b=$910; 540-10b+20b=$910; 540+10b=$910; 10b=910-540; 10b=370; b=37; so there are 37 20's and since a+b=54, a+37=54; a=54-37=17; a=17, so 17 10's. So lets check. $740+$170=$910. David Reply . A cashier has 54 bills, all of which are$10 or $20 bills. The total value of the money is$910. How many of each type of bill does the cashier have?
whats the coefficient of 17x
the solution says it 14 but how i thought it would be 17 im i right or wrong is the exercise wrong
Dwayne
17
Melissa
wow the exercise told me 17x solution is 14x lmao
Dwayne
thank you
Dwayne
A private jet can fly 1,210 miles against a 25 mph headwind in the same amount of time it can fly 1,694 miles with a 25 mph tailwind. Find the speed of the jet
Washing his dad’s car alone, eight-year-old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take the Levi’s dad to wash the car by himself?
Ethan and Leo start riding their bikes at the opposite ends of a 65-mile bike path. After Ethan has ridden 1.5 hours and Leo has ridden 2 hours, they meet on the path. Ethan’s speed is 6 miles per hour faster than Leo’s speed. Find the speed of the two bikers.
Nathan walked on an asphalt pathway for 12 miles. He walked the 12 miles back to his car on a gravel road through the forest. On the asphalt he walked 2 miles per hour faster than on the gravel. The walk on the gravel took one hour longer than the walk on the asphalt. How fast did he walk on the gravel?
Mckenzie
Nancy took a 3 hour drive. She went 50 miles before she got caught in a storm. Then she drove 68 miles at 9 mph less than she had driven when the weather was good. What was her speed driving in the storm?
Mr Hernaez runs his car at a regular speed of 50 kph and Mr Ranola at 36 kph. They started at the same place at 5:30 am and took opposite directions. At what time were they 129 km apart?
90 minutes
Melody wants to sell bags of mixed candy at her lemonade stand. She will mix chocolate pieces that cost $4.89 per bag with peanut butter pieces that cost$3.79 per bag to get a total of twenty-five bags of mixed candy. Melody wants the bags of mixed candy to cost her $4.23 a bag to make. How many bags of chocolate pieces and how many bags of peanut butter pieces should she use? Jake Reply enrique borrowed$23,500 to buy a car he pays his uncle 2% interest on the $4,500 he borrowed from him and he pays the bank 11.5% interest on the rest. what average interest rate does he pay on the total$23,500
13.5
Pervaiz
Amber wants to put tiles on the backsplash of her kitchen counters. She will need 36 square feet of tiles. She will use basic tiles that cost $8 per square foot and decorator tiles that cost$20 per square foot. How many square feet of each tile should she use so that the overall cost of the backsplash will be \$10 per square foot?
The equation P=28+2.54w models the relation between the amount of Randy’s monthly water bill payment, P, in dollars, and the number of units of water, w, used. Find the payment for a month when Randy used 15 units of water.
Bridget
help me understand graphs
what kind of graphs?
bruce
function f(x) to find each value
Marlene
I am in algebra 1. Can anyone give me any ideas to help me learn this stuff. Teacher and tutor not helping much.
Marlene
Given f(x)=2x+2, find f(2) so you replace the x with the 2, f(2)=2(2)+2, which is f(2)=6
Melissa
if they say find f(5) then the answer would be f(5)=12
Melissa
I need you to help me Melissa. Wish I can show you my homework
Marlene
How is f(1) =0 I am really confused
Marlene
what's the formula given? f(x)=?
Melissa
It shows a graph that I wish I could send photo of to you on here
Marlene
Which problem specifically?
Melissa
which problem?
Melissa
I don't know any to be honest. But whatever you can help me with for I can practice will help
Marlene
I got it. sorry, was out and about. I'll look at it now.
Melissa
Thank you. I appreciate it because my teacher assumes I know this. My teacher before him never went over this and several other things.
Marlene
I just responded.
Melissa
Thank you
Marlene
-65r to the 4th power-50r cubed-15r squared+8r+23 ÷ 5r
Rich
write in this form a/b answer should be in the simplest form 5%
convert to decimal 9/11
August
0.81818
Rich
5/100 = .05 but Rich is right that 9/11 = .81818
Melissa