# 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?

how did you get the value of 2000N.What calculations are needed to arrive at it
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Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike? Got questions? Get instant answers now!
how do u solve that question
Seera
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
Seera
Speed=distance ÷ time
Tremayne
x-3y =1; 3x-2y+4=0 graph
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