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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: ⓐ $\sqrt[3]{54}-\sqrt[3]{16}$ ⓑ $\sqrt[4]{48}+\sqrt[4]{243}$ .
Simplify: ⓐ $\sqrt[3]{192}-\sqrt[3]{81}$ ⓑ $\sqrt[4]{32}+\sqrt[4]{512}$ .
ⓐ $\sqrt[3]{3}$ ⓑ $6\sqrt[4]{2}$
Simplify: ⓐ $\sqrt[3]{108}-\sqrt[3]{250}$ ⓑ $\sqrt[5]{64}+\sqrt[5]{486}$ .
ⓐ $\text{\u2212}\sqrt[3]{2}$ ⓑ $5\sqrt[5]{2}$
Simplify: ⓐ $\sqrt[3]{24{x}^{4}}-\sqrt[3]{\mathrm{-81}{x}^{7}}$ ⓑ $\sqrt[4]{162{y}^{9}}+\sqrt[4]{516{y}^{5}}$ .
Simplify: ⓐ $\sqrt[3]{32{y}^{5}}-\sqrt[3]{\mathrm{-108}{y}^{8}}$ ⓑ $\sqrt[4]{243{r}^{11}}+\sqrt[4]{768{r}^{10}}$ .
ⓐ $2y\sqrt[3]{4{y}^{2}}+3{y}^{2}\sqrt[3]{4{y}^{2}}$ ⓑ $3{r}^{2}\sqrt[4]{3{r}^{3}}+4{r}^{2}\sqrt[4]{3{r}^{2}}$
Simplify: ⓐ $\sqrt[3]{40{z}^{7}}-\sqrt[3]{\mathrm{-135}{z}^{4}}$ ⓑ $\sqrt[4]{80{s}^{13}}+\sqrt[4]{1280{s}^{6}}$ .
ⓐ $2{z}^{2}\sqrt[3]{5z}+3z\sqrt[3]{5z}$ ⓑ $2\left|{s}^{3}\right|\sqrt[4]{5s}+4\left|s\right|\sqrt[4]{5s}$
Access these online resources for additional instruction and practice with simplifying higher roots.
Simplify Expressions with Higher Roots
In the following exercises, simplify.
ⓐ
$\sqrt[3]{216}$
ⓑ
$\sqrt[4]{256}$
ⓒ
$\sqrt[5]{32}$
ⓐ
$\sqrt[3]{27}$
ⓑ
$\sqrt[4]{16}$
ⓒ
$\sqrt[5]{243}$
ⓐ $3$ ⓑ $2$ ⓒ $3$
ⓐ
$\sqrt[3]{512}$
ⓑ
$\sqrt[4]{81}$
ⓒ
$\sqrt[5]{1}$
ⓐ
$\sqrt[3]{125}$
ⓑ
$\sqrt[4]{1296}$
ⓒ
$\sqrt[5]{1024}$
ⓐ $5$ ⓑ $6$ ⓒ $4$
ⓐ
$\sqrt[3]{\mathrm{-8}}$
ⓑ
$\sqrt[4]{\mathrm{-81}}$
ⓒ
$\sqrt[5]{\mathrm{-32}}$
ⓐ
$\sqrt[3]{\mathrm{-64}}$
ⓑ
$\sqrt[4]{\mathrm{-16}}$
ⓒ
$\sqrt[5]{\mathrm{-243}}$
ⓐ $\mathrm{-4}$ ⓑ $\text{not real}$ ⓒ $\mathrm{-3}$
ⓐ
$\sqrt[3]{\mathrm{-125}}$
ⓑ
$\sqrt[4]{\mathrm{-1296}}$
ⓒ
$\sqrt[5]{\mathrm{-1024}}$
ⓐ
$\sqrt[3]{\mathrm{-512}}$
ⓑ
$\sqrt[4]{\mathrm{-81}}$
ⓒ
$\sqrt[5]{\mathrm{-1}}$
ⓐ $\mathrm{-8}$ ⓑ not a real number ⓒ $\mathrm{-1}$
ⓐ
$\sqrt[5]{{u}^{5}}$
ⓑ
$\sqrt[8]{{v}^{8}}$
ⓐ
$\sqrt[4]{{y}^{4}}$
ⓑ
$\sqrt[7]{{m}^{7}}$
ⓐ
$\sqrt[8]{{k}^{8}}$
ⓑ
$\sqrt[6]{{p}^{6}}$
ⓐ $\left|k\right|$ ⓑ $\left|p\right|$
ⓐ
$\sqrt[3]{{x}^{9}}$
ⓑ
$\sqrt[4]{{y}^{12}}$
ⓐ
$\sqrt[5]{{a}^{10}}$
ⓑ
$\sqrt[3]{{b}^{27}}$
ⓐ ${a}^{2}$ ⓑ ${b}^{9}$
ⓐ
$\sqrt[4]{{m}^{8}}$
ⓑ
$\sqrt[5]{{n}^{20}}$
ⓐ
$\sqrt[6]{{r}^{12}}$
ⓑ
$\sqrt[3]{{s}^{30}}$
ⓐ ${r}^{2}$ ⓑ ${s}^{10}$
ⓐ
$\sqrt[4]{16{x}^{8}}$
ⓑ
$\sqrt[6]{64{y}^{12}}$
ⓐ
$\sqrt[3]{\mathrm{-8}{c}^{9}}$
ⓑ
$\sqrt[3]{125{d}^{15}}$
ⓐ $\mathrm{-2}{c}^{3}$ ⓑ $5{d}^{5}$
ⓐ
$\sqrt[3]{216{a}^{6}}$
ⓑ
$\sqrt[5]{32{b}^{20}}$
ⓐ
$\sqrt[7]{128{r}^{14}}$
ⓑ
$\sqrt[4]{81{s}^{24}}$
ⓐ $2{r}^{2}$ ⓑ $3{s}^{6}$
Use the Product Property to Simplify Expressions with Higher Roots
In the following exercises, simplify.
ⓐ $\sqrt[3]{{r}^{5}}$ ⓑ $\sqrt[4]{{s}^{10}}$
ⓐ $\sqrt[5]{{u}^{7}}$ ⓑ $\sqrt[6]{{v}^{11}}$
ⓐ $u\sqrt[5]{{u}^{2}}$ ⓑ $v\sqrt[6]{{v}^{5}}$
ⓐ $\sqrt[4]{{m}^{5}}$ ⓑ $\sqrt[8]{{n}^{10}}$
ⓐ $\sqrt[5]{{p}^{8}}$ ⓑ $\sqrt[3]{{q}^{8}}$
ⓐ $p\sqrt[5]{{p}^{3}}$ ⓑ ${q}^{2}\sqrt[3]{{q}^{2}}$
ⓐ $\sqrt[4]{32}$ ⓑ $\sqrt[5]{64}$
ⓐ $\sqrt[3]{625}$ ⓑ $\sqrt[6]{128}$
ⓐ $5\sqrt[3]{5}$ ⓑ $2\sqrt[6]{2}$
ⓐ $\sqrt[5]{64}$ ⓑ $\sqrt[3]{256}$
ⓐ $\sqrt[4]{3125}$ ⓑ $\sqrt[3]{81}$
ⓐ $5\sqrt[4]{5}$ ⓑ $3\sqrt[3]{3}$
ⓐ $\sqrt[3]{108{x}^{5}}$ ⓑ $\sqrt[4]{48{y}^{6}}$
ⓐ $\sqrt[5]{96{a}^{7}}$ ⓑ $\sqrt[3]{375{b}^{4}}$
ⓐ $2a\sqrt[5]{3{a}^{2}}$ ⓑ $5b\sqrt[3]{3b}$
ⓐ $\sqrt[4]{405{m}^{10}}$ ⓑ $\sqrt[5]{160{n}^{8}}$
ⓐ $\sqrt[3]{512{p}^{5}}$ ⓑ $\sqrt[4]{324{q}^{7}}$
ⓐ $8p\sqrt[3]{{p}^{2}}$ ⓑ $3q\sqrt[4]{4{q}^{3}}$
ⓐ $\sqrt[3]{\mathrm{-864}}$ ⓑ $\sqrt[4]{\mathrm{-256}}$
ⓐ $\sqrt[5]{\mathrm{-486}}$ ⓑ $\sqrt[6]{\mathrm{-64}}$
ⓐ $\mathrm{-3}\sqrt[5]{2}$ ⓑ $\text{not real}$
ⓐ $\sqrt[5]{\mathrm{-32}}$ ⓑ $\sqrt[8]{\mathrm{-1}}$
ⓐ $\sqrt[3]{\mathrm{-8}}$ ⓑ $\sqrt[4]{\mathrm{-16}}$
ⓐ $\mathrm{-2}$ ⓑ $\text{not real}$
Use the Quotient Property to Simplify Expressions with Higher Roots
In the following exercises, simplify.
ⓐ $\sqrt[3]{\frac{{p}^{11}}{{p}^{2}}}$ ⓑ $\sqrt[4]{\frac{{q}^{17}}{{q}^{13}}}$
ⓐ $\sqrt[5]{\frac{{d}^{12}}{{d}^{7}}}$ ⓑ $\sqrt[8]{\frac{{m}^{12}}{{m}^{4}}}$
ⓐ $d$ ⓑ $\left|m\right|$
ⓐ $\sqrt[5]{\frac{{u}^{21}}{{u}^{11}}}$ ⓑ $\sqrt[6]{\frac{{v}^{30}}{{v}^{12}}}$
ⓐ $\sqrt[3]{\frac{{r}^{14}}{{r}^{5}}}$ ⓑ $\sqrt[4]{\frac{{c}^{21}}{{c}^{9}}}$
ⓐ ${r}^{2}$ ⓑ $\left|{c}^{3}\right|$
ⓐ $\frac{\sqrt[4]{64}}{\sqrt[4]{2}}$ ⓑ $\frac{\sqrt[5]{128{x}^{8}}}{\sqrt[5]{2{x}^{2}}}$
ⓐ $\frac{\sqrt[3]{\mathrm{-625}}}{\sqrt[3]{5}}$ ⓑ $\frac{\sqrt[4]{80{m}^{7}}}{\sqrt[4]{5m}}$
ⓐ $\mathrm{-5}$ ⓑ $4m\sqrt[4]{{m}^{2}}$
ⓐ $\sqrt[3]{\frac{1050}{2}}$ ⓑ $\sqrt[4]{\frac{486{y}^{9}}{2{y}^{3}}}$
ⓐ $\sqrt[3]{\frac{162}{6}}$ ⓑ $\sqrt[4]{\frac{160{r}^{10}}{5{r}^{3}}}$
ⓐ $3\sqrt[3]{6}$ ⓑ $2\left|r\right|\sqrt[4]{2{r}^{3}}$
ⓐ $\sqrt[3]{\frac{54{a}^{8}}{{b}^{3}}}$ ⓑ $\sqrt[4]{\frac{64{c}^{5}}{{d}^{2}}}$
ⓐ $\sqrt[5]{\frac{96{r}^{11}}{{s}^{3}}}$ ⓑ $\sqrt[6]{\frac{128{u}^{7}}{{v}^{3}}}$
ⓐ $\frac{2{r}^{2}\sqrt[5]{3r}}{{s}^{3}}$ ⓑ $\frac{2{u}^{3}\sqrt[6]{2uv3}}{v}$
ⓐ $\sqrt[3]{\frac{81{s}^{8}}{{t}^{3}}}$ ⓑ $\sqrt[4]{\frac{64{p}^{15}}{{q}^{12}}}$
ⓐ $\sqrt[3]{\frac{625{u}^{10}}{{v}^{3}}}$ ⓑ $\sqrt[4]{\frac{729{c}^{21}}{{d}^{8}}}$
ⓐ $\frac{5{u}^{3}\sqrt[3]{5u}}{v}$ ⓑ $\frac{3{c}^{5}\sqrt[4]{9c}}{{d}^{2}}$
Add and Subtract Higher Roots
In the following exercises, simplify.
ⓐ
$\sqrt[7]{8p}+\sqrt[7]{8p}$
ⓑ
$3\sqrt[3]{25}-\sqrt[3]{25}$
ⓐ
$\sqrt[3]{15q}+\sqrt[3]{15q}$
ⓑ
$2\sqrt[4]{27}-6\sqrt[4]{27}$
ⓐ $2\sqrt[3]{15q}$ ⓑ $\mathrm{-4}\sqrt[4]{27}$
ⓐ
$3\sqrt[5]{9x}+7\sqrt[5]{9x}$
ⓑ
$8\sqrt[7]{3q}-2\sqrt[7]{3q}$
ⓐ
$\sqrt[3]{81}-\sqrt[3]{192}$
ⓑ
$\sqrt[4]{512}-\sqrt[4]{32}$
ⓐ
$\sqrt[3]{250}-\sqrt[3]{54}$
ⓑ
$\sqrt[4]{243}-\sqrt[4]{1875}$
ⓐ $5\sqrt[3]{5}-3\sqrt[3]{2}$ ⓑ $\mathrm{-2}\sqrt[4]{3}$
ⓐ
$\sqrt[3]{128}+\sqrt[3]{250}$
ⓑ
$\sqrt[5]{729}+\sqrt[5]{96}$
ⓐ
$\sqrt[4]{243}+\sqrt[4]{1250}$
ⓑ
$\sqrt[3]{2000}+\sqrt[3]{54}$
ⓐ $3\sqrt[4]{3}+5\sqrt[4]{2}$ ⓑ $13\sqrt[3]{2}$
ⓐ
$\sqrt[3]{64{a}^{10}}-\sqrt[3]{\mathrm{-216}{a}^{12}}$
ⓑ
$\sqrt[4]{486{u}^{7}}+\sqrt[4]{768{u}^{3}}$
ⓐ
$\sqrt[3]{80{b}^{5}}-\sqrt[3]{\mathrm{-270}{b}^{3}}$
ⓑ
$\sqrt[4]{160{v}^{10}}-\sqrt[4]{1280{v}^{3}}$
ⓐ $2b\sqrt[3]{10{b}^{2}}+3b\sqrt[3]{10}$ ⓑ $2{v}^{2}\sqrt[4]{10{v}^{2}}-4\sqrt[4]{5{v}^{3}}$
Mixed Practice
In the following exercises, simplify.
$\sqrt[4]{16}$
$\sqrt[3]{{a}^{3}}$
$\sqrt[3]{\mathrm{-8}{c}^{9}}$
$\sqrt[3]{{r}^{5}}$
$\sqrt[3]{108{x}^{5}}$
$\sqrt[5]{\mathrm{-486}}$
$\frac{\sqrt[4]{64}}{\sqrt[4]{2}}$
$\frac{\sqrt[5]{128{x}^{8}}}{\sqrt[5]{2{x}^{2}}}$
$2x\sqrt[5]{2x}$
$\sqrt[5]{\frac{96{r}^{11}}{{s}^{3}}}$
$\sqrt[6]{\frac{128{u}^{7}}{{v}^{3}}}$
$\frac{2{u}^{3}\sqrt[6]{2uv3}}{v}$
$\sqrt[3]{81}-\sqrt[3]{192}$
$\sqrt[3]{64{a}^{10}}-\sqrt[3]{\mathrm{-216}{a}^{12}}$
$\sqrt[4]{486{u}^{7}}+\sqrt[4]{768{u}^{3}}$
$3u\sqrt[4]{6{u}^{3}}+4\sqrt[4]{3{u}^{3}}$
Population growth The expression $10\xb7{x}^{n}$ models the growth of a mold population after $n$ generations. There were 10 spores at the start, and each had $x$ offspring. So $10\xb7{x}^{n}$ is the number of offspring at the fifth generation. At the fifth generation there were 10,240 offspring. Simplify the expression $\sqrt[5]{\frac{\mathrm{10,240}}{10}}$ to determine the number of offspring of each spore.
Spread of a virus The expression $3\xb7{x}^{n}$ 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 $3\xb7{x}^{4}$ 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.
$5$
Explain how you know that $\sqrt[5]{{x}^{10}}={x}^{2}$ .
Explain why $\sqrt[4]{\mathrm{-64}}$ is not a real number but $\sqrt[3]{\mathrm{-64}}$ is.
Answers may vary.
ⓐ 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?
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