# 9.7 Higher roots  (Page 4/8)

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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 to square
easiest way to find the square root of a large number?
Jackie
the accompanying figure shows known flow rates of hydrocarbons into and out of a network of pipes at an oil refinery set up a linear system whose solution provides the unknown flow rates (b) solve the system for the unknown flow rates (c) find the flow rates and directions of flow if x4=50and x6=0
What is observation
I'm confused by the question. Can you describe or explain the math question it pertains to?
Melissa
there is no math to it because all you use is your vision or gaze to the sorrounding areas
Cesarp
Teegan likes to play golf. He has budgeted $60 next month for the driving range. It costs him$10.55 for a bucket of balls each time he goes. What is the maximum number of times he can go to the driving range next month?
5 times max
Anton
Felecia left her home to visit her daughter, driving 45mph. Her husband waited for the dog sitter to arrive and left home 20 minutes, or 1/3 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her?
35 min
Debra
Carmen wants to tile the floor of his house. He will need 1,000 square feet of tile. He will do most of the floor with a tile that costs $1.50 per square foot, but also wants to use an accent tile that costs$9.00 per square foot. How many square feet of each tile should he plan to use if he wants the overall cost to be $3 per square foot? Parker Reply what you wanna get Cesar 800 sq. ft @$1.50 & 200 sq. ft @ $9.00 Marco Geneva treated her parents to dinner at their favorite restaurant. The bill was$74.25. Geneva wants to leave 16 % of the total - bill as a tip. How much should the tip be?
74.25 × .16 then get the total and that will be your tip
David
$74.25 x 0.16 =$11.88 total bill: $74.25 +$11.88 = $86.13 ericka yes and tip 16% will be$11.88
David
what is the shorter way to do it
Priam has dimes and pennies in a cup holder in his car. The total value of the coins is $4.21. The number of dimes is three less than four times the number of pennies. How many dimes and how many pennies are in the cup? Alexandra Reply Uno de los ángulos suplementario es 4° más que 1/3 del otro ángulo encuentra las medidas de cada uno de los angulos Enith Reply June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold? Alexandra Reply I hope this is correct, x=cooler 1 5x=cooler 2 x + 5x = 48 6x=48 ×=8 gallons 5×=40 gallons ericka Priam has pennies and dimes in a cup holder in his car. The total value of the coins is$4.21 . The number of dimes is three less than four times the number of pennies. How many pennies and how many dimes are in the cup?
Arnold invested $64,000 some at 5.5% interest and the rest at 9% interest how much did he invest at each rate if he received$4500 in interest in one year
List five positive thoughts you can say to yourself that will help youapproachwordproblemswith a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often.
Avery and Caden have saved \$27,000 towards a down payment on a house. They want to keep some of the money in a bank account that pays 2.4% annual interest and the rest in a stock fund that pays 7.2% annual interest. How much should they put into each account so that they earn 6% interest per year?
324.00
Irene
1.2% of 27.000
Irene
i did 2.4%-7.2% i got 1.2%
Irene
I have 6% of 27000 = 1620 so we need to solve 2.4x +7.2y =1620
Catherine
I think Catherine is on the right track. Solve for x and y.
Scott
next bit : x=(1620-7.2y)/2.4 y=(1620-2.4x)/7.2 I think we can then put the expression on the right hand side of the "x=" into the second equation. 2.4x in the second equation can be rewritten as 2.4(rhs of first equation) I write this out tidy and get back to you...
Catherine