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Higher Roots

Simplify expressions with higher roots

Simplify Expressions with Higher Roots

Up to now, in this chapter we have worked with squares and square roots. We will now extend our work to include higher powers and higher roots.

Let’s review some vocabulary first.

\[\begin{array}{lll}\text{We write:} & & \ \text{We say:} \\ {n}^{2} & & \ n\ \text{squared} \\ {n}^{3} & & \ n\ \text{cubed} \\ {n}^{4} & & \ n\ \text{to the fourth} \\ {n}^{5} & & \ n\ \text{to the fifth}\end{array}\]

The terms ‘squared’ and ‘cubed’ come from the formulas for area of a square and volume of a cube.

It will be helpful to have a table of the powers of the integers from \(-5\ \text{to}\ 5\). See .

Notice the signs in . All powers of positive numbers are positive, of course. But when we have a negative number, the even powers are positive and the odd powers are negative. We’ll copy the row with the powers of \(-2\) below to help you see this.

Earlier in this chapter we defined the square root of a number.

\[\text{If}\ {n}^{2}=m,\ \text{then}\ n\ \text{is a square root of}\ m.\]

And we have used the notation \(\sqrt{m}\) to denote the principal square root. So \(\sqrt{m}\ge 0\) always.

\[\begin{array}{lllllll}{4}^{3} & = & 64 & & \ \sqrt[3]{64} & = & 4 \\ {3}^{4} & = & 81 & & \ \sqrt[4]{81} & = & 3 \\ {(-2)}^{5} & = & -32 & & \ \sqrt[5]{-32} & = & -2\end{array}\]

Example

Try it.

Simplify: ⓐ \(\sqrt[3]{8}\) ⓑ \(\sqrt[4]{81}\) ⓒ \(\sqrt[5]{32}\).

Solution

ⓐ

	\(\sqrt[3]{8}\)
Since \({(2)}^{3}=8.\)	\(2\)

ⓑ

	\(\sqrt[4]{81}\)
Since \({(3)}^{4}=81.\)	\(3\)

ⓒ

	\(\sqrt[5]{32}\)
Since \({(2)}^{5}=32.\)	\(2\)

Example

Try it.

Simplify: ⓐ \(\sqrt[3]{-64}\) ⓑ \(\sqrt[4]{-16}\) ⓒ \(\sqrt[5]{-243}\).

Solution

ⓐ

	\(\sqrt[3]{-64}\)
Since \({(-4)}^{3}=-64.\)	\(-4\)

ⓑ

	\(\sqrt[4]{-16}\)
Think, \({(?)}^{4}=-16.\) No real number raised to the fourth power is negative.	Not a real number.

ⓒ

	\(\sqrt[5]{-243}\)
Since \({(-3)}^{5}=-243.\)	\(-3\)

\[\begin{array}{lll}{(-5)}^{4}=625 & & \ \sqrt[4]{625}=5\end{array}\]

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Use the Product Property to Simplify Expressions with Higher Roots

We will simplify expressions with higher roots in much the same way as we simplified expressions with square roots. An nth root is considered simplified if it has no factors of \({m}^{n}\).

We will generalize the Product Property of Square Roots to include any integer root \(n\ge 2\).

Example

Try it.

Simplify: ⓐ \(\sqrt[3]{{x}^{4}}\) ⓑ \(\sqrt[4]{{x}^{7}}\).

Solution

ⓐ

	\(\sqrt[3]{{x}^{4}}\)
Rewrite the radicand as a product using the largest perfect cube factor.	\(\sqrt[3]{{x}^{3}\cdot x}\)
Rewrite the radical as the product of two radicals.	\(\sqrt[3]{{x}^{3}}\cdot \sqrt[3]{x}\)
Simplify.	\(x\sqrt[3]{x}\)

ⓑ

	\(\sqrt[4]{{x}^{7}}\)
Rewrite the radicand as a product using the greatest perfect fourth power factor.	\(\sqrt[4]{{x}^{4}\cdot {x}^{3}}\)
Rewrite the radical as the product of two radicals.	\(\sqrt[4]{{x}^{4}}\cdot \sqrt[4]{{x}^{3}}\)
Simplify.	\(\|x\|\sqrt[4]{{x}^{3}}\)

Example

Try it.

Simplify: ⓐ \(\sqrt[3]{16}\) ⓑ \(\sqrt[4]{243}\).

Solution

ⓐ

	\(\sqrt[3]{16}\)
	\(\sqrt[3]{{2}^{4}}\)
Rewrite the radicand as a product using the greatest perfect cube factor.	\(\sqrt[3]{{2}^{3}\cdot 2}\)
Rewrite the radical as the product of two radicals.	\(\sqrt[3]{{2}^{3}}\cdot \sqrt[3]{2}\)
Simplify.	\(2\sqrt[3]{2}\)

ⓑ

	\(\sqrt[4]{243}\)
	\(\sqrt[4]{{3}^{5}}\)
Rewrite the radicand as a product using the greatest perfect fourth power factor.	\(\sqrt[4]{{3}^{4}\cdot 3}\)
Rewrite the radical as the product of two radicals.	\(\sqrt[4]{{3}^{4}}\cdot \sqrt[4]{3}\)
Simplify.	\(3\sqrt[4]{3}\)

Don’t forget to use the absolute value signs when taking an even root of an expression with a variable in the radical.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Use the Quotient Property to Simplify Expressions with Higher Roots

We can simplify higher roots with quotients in the same way we simplified square roots. First we simplify any fractions inside the radical.

Example

Try it.

Simplify: ⓐ \(\sqrt[3]{\frac{{a}^{8}}{{a}^{5}}}\) ⓑ \(\sqrt[4]{\frac{{a}^{10}}{{a}^{2}}}\).

Solution

ⓐ

	\(\sqrt[3]{\frac{{a}^{8}}{{a}^{5}}}\)
Simplify the fraction under the radical first.	\(\sqrt[3]{{a}^{3}}\)
Simplify.	\(a\)

ⓑ

	\(\sqrt[4]{\frac{{a}^{10}}{{a}^{2}}}\)
Simplify the fraction under the radical first.	\(\sqrt[4]{{a}^{8}}\)
Rewrite the radicand using perfect fourth power factors.	\(\sqrt[4]{{({a}^{2})}^{4}}\)
Simplify.	\({a}^{2}\)

Previously, we used the Quotient Property ‘in reverse’ to simplify square roots. Now we will generalize the formula to include higher roots.

Example

Try it.

Simplify: ⓐ \(\frac{\sqrt[3]{-108}}{\sqrt[3]{2}}\) ⓑ \(\frac{\sqrt[4]{96{x}^{7}}}{\sqrt[4]{3{x}^{2}}}\).

Solution

ⓐ

	\(\frac{\sqrt[3]{-108}}{\sqrt[3]{2}}\)
Neither radicand is a perfect cube, so use the Quotient Property to write as one radical.	\(\sqrt[3]{\frac{-108}{2}}\)
Simplify the fraction under the radical.	\(\sqrt[3]{-54}\)
Rewrite the radicand as a product using perfect cube factors.	\(\sqrt[3]{{(-3)}^{3}\cdot 2}\)
Rewrite the radical as the product of two radicals.	\(\sqrt[3]{{(-3)}^{3}}\cdot \sqrt[3]{2}\)
Simplify.	\(-3\sqrt[3]{2}\)

ⓑ

	\(\frac{\sqrt[4]{96{x}^{7}}}{\sqrt[4]{3{x}^{2}}}\)
Neither radicand is a perfect fourth power, so use the Quotient Property to write as one radical.	\(\sqrt[4]{\frac{96{x}^{7}}{3{x}^{2}}}\)
Simplify the fraction under the radical.	\(\sqrt[4]{32{x}^{5}}\)
Rewrite the radicand as a product using perfect fourth power factors.	\(\sqrt[4]{{2}^{4}{x}^{4}\cdot 2x}\)
Rewrite the radical as the product of two radicals.	\(\sqrt[4]{{(2x)}^{4}}\cdot \sqrt[4]{2x}\)
Simplify.	\(2\|x\|\sqrt[4]{2x}\)

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Add and Subtract Higher Roots

We can add and subtract higher roots like we added and subtracted square roots. First we provide a formal definition of like radicals.

Like radicals have the same index and the same radicand.

\(9\sqrt[4]{42x}\) and \(-2\sqrt[4]{42x}\) are like radicals.
\(5\sqrt[3]{125x}\) and \(6\sqrt[3]{125y}\) are not like radicals. The radicands are different.
\(2\sqrt[5]{1000q}\) and \(-4\sqrt[4]{1000q}\) are not like radicals. The indices are different.

We add and subtract like radicals in the same way we add and subtract like terms. We can add \(9\sqrt[4]{42x}+(-2\sqrt[4]{42x})\) and the result is \(7\sqrt[4]{42x}\).

Example

Try it.

Simplify: ⓐ \(\sqrt[3]{4x}+\sqrt[3]{4x}\) ⓑ \(4\sqrt[4]{8}-2\sqrt[4]{8}\).

Solution

ⓐ

	\(\sqrt[3]{4x}+\sqrt[3]{4x}\)
The radicals are like, so we add the coefficients.	\(2\sqrt[3]{4x}\)

ⓑ

	\(4\sqrt[4]{8}-2\sqrt[4]{8}\)
The radicals are like, so we subtract the coefficients.	\(2\sqrt[4]{8}\)

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.

Example

Try it.

Simplify: ⓐ \(\sqrt[3]{54}-\sqrt[3]{16}\) ⓑ \(\sqrt[4]{48}+\sqrt[4]{243}\).

Solution

ⓐ

	\(\sqrt[3]{54}-\sqrt[3]{16}\)
Rewrite each radicand using perfect cube factors.	\(\sqrt[3]{27}\cdot \sqrt[3]{2}-\sqrt[3]{8}\cdot \sqrt[3]{2}\)
Rewrite the perfect cubes.	\(\sqrt[3]{{(3)}^{3}}\ \sqrt[3]{2}-\sqrt[3]{{(2)}^{3}}\ \sqrt[3]{2}\)
Simplify the radicals where possible.	\(3\sqrt[3]{2}-2\sqrt[3]{2}\)
Combine like radicals.	\(\sqrt[3]{2}\)

ⓑ

	\(\sqrt[4]{48}+\sqrt[4]{243}\)
Rewrite using perfect fourth power factors.	\(\sqrt[4]{16}\cdot \sqrt[4]{3}+\sqrt[4]{81}\cdot \sqrt[4]{3}\)
Rewrite the perfect fourth powers.	\(\sqrt[4]{{(2)}^{4}}\ \sqrt[4]{3}+\sqrt[4]{{(3)}^{4}}\ \sqrt[4]{3}\)
Simplify the radicals where possible.	\(2\sqrt[4]{3}+3\sqrt[4]{3}\)
Combine like radicals.	\(5\sqrt[4]{3}\)

Condensed — the full section is in OpenStax Elementary Algebra 2e.

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 nth Roots
\[\sqrt[n]{ab}=\sqrt[n]{a}\cdot \sqrt[n]{b}\ \text{and}\ \sqrt[n]{a}\cdot \sqrt[n]{b}=\sqrt[n]{ab}\]
Quotient Property of nth Roots
\[\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}\ \text{and}\ \frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}\]
To combine like radicals, simply add or subtract the coefficients while keeping the radical the same.

Higher Roots

Simplify Expressions with Higher Roots

In the following exercises, simplify.

Try it.

ⓐ \(\sqrt[3]{216}\) ⓑ \(\sqrt[4]{256}\) ⓒ \(\sqrt[5]{32}\)

Try it.

ⓐ \(\sqrt[3]{27}\) ⓑ \(\sqrt[4]{16}\) ⓒ \(\sqrt[5]{243}\)

Solution

ⓐ \(3\) ⓑ \(2\) ⓒ \(3\)

Try it.

ⓐ \(\sqrt[3]{512}\) ⓑ \(\sqrt[4]{81}\) ⓒ \(\sqrt[5]{1}\)

Try it.

ⓐ \(\sqrt[3]{125}\) ⓑ \(\sqrt[4]{1296}\) ⓒ \(\sqrt[5]{1024}\)

Solution

ⓐ \(5\) ⓑ \(6\) ⓒ \(4\)

Try it.

ⓐ \(\sqrt[3]{-8}\) ⓑ \(\sqrt[4]{-81}\) ⓒ \(\sqrt[5]{-32}\)

Try it.

ⓐ \(\sqrt[3]{-64}\) ⓑ \(\sqrt[4]{-16}\) ⓒ \(\sqrt[5]{-243}\)

Solution

ⓐ \(-4\) ⓑ \(\text{not real}\) ⓒ \(-3\)

Try it.

ⓐ \(\sqrt[3]{-125}\) ⓑ \(\sqrt[4]{-1296}\) ⓒ \(\sqrt[5]{-1024}\)

Try it.

ⓐ \(\sqrt[3]{-512}\) ⓑ \(\sqrt[4]{-81}\) ⓒ \(\sqrt[5]{-1}\)

Solution

ⓐ \(-8\) ⓑ not a real number ⓒ \(-1\)

Try it.

ⓐ \(\sqrt[5]{{u}^{5}}\) ⓑ \(\sqrt[8]{{v}^{8}}\)

Try it.

ⓐ \(\sqrt[3]{{a}^{3}}\)
ⓑ

Solution

ⓐ \(a\) ⓑ \(|b|\)

Try it.

ⓐ \(\sqrt[4]{{y}^{4}}\) ⓑ \(\sqrt[7]{{m}^{7}}\)

Try it.

ⓐ \(\sqrt[8]{{k}^{8}}\) ⓑ \(\sqrt[6]{{p}^{6}}\)

Solution

ⓐ \(|k|\) ⓑ \(|p|\)

Try it.

ⓐ \(\sqrt[3]{{x}^{9}}\) ⓑ \(\sqrt[4]{{y}^{12}}\)

Try it.

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

Solution

ⓐ \({a}^{2}\) ⓑ \({b}^{9}\)

Try it.

ⓐ \(\sqrt[4]{{m}^{8}}\) ⓑ \(\sqrt[5]{{n}^{20}}\)

Try it.

ⓐ \(\sqrt[6]{{r}^{12}}\) ⓑ \(\sqrt[3]{{s}^{30}}\)

Solution

ⓐ \({r}^{2}\) ⓑ \({s}^{10}\)

Try it.

ⓐ \(\sqrt[4]{16{x}^{8}}\) ⓑ \(\sqrt[6]{64{y}^{12}}\)

Try it.

ⓐ \(\sqrt[3]{-8{c}^{9}}\) ⓑ \(\sqrt[3]{125{d}^{15}}\)

Solution

ⓐ \(-2{c}^{3}\) ⓑ \(5{d}^{5}\)

Try it.

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

Try it.

ⓐ \(\sqrt[7]{128{r}^{14}}\) ⓑ \(\sqrt[4]{81{s}^{24}}\)

Solution

ⓐ \(2{r}^{2}\) ⓑ \(3{s}^{6}\)

Use the Product Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

Try it.

ⓐ \(\sqrt[3]{{r}^{5}}\) ⓑ \(\sqrt[4]{{s}^{10}}\)

Try it.

ⓐ \(\sqrt[5]{{u}^{7}}\) ⓑ \(\sqrt[6]{{v}^{11}}\)

Solution

ⓐ \(u\sqrt[5]{{u}^{2}}\) ⓑ \(\left|v\right|\sqrt[6]{{v}^{5}}\)

Try it.

ⓐ \(\sqrt[4]{{m}^{5}}\) ⓑ \(\sqrt[8]{{n}^{10}}\)

Try it.

ⓐ \(\sqrt[5]{{p}^{8}}\) ⓑ \(\sqrt[3]{{q}^{8}}\)

Solution

ⓐ \(p\sqrt[5]{{p}^{3}}\) ⓑ \({q}^{2}\sqrt[3]{{q}^{2}}\)

Try it.

ⓐ \(\sqrt[4]{32}\) ⓑ \(\sqrt[7]{64}\)

Try it.

ⓐ \(\sqrt[3]{625}\) ⓑ \(\sqrt[6]{128}\)

Solution

ⓐ \(5\sqrt[3]{5}\) ⓑ \(2\sqrt[6]{2}\)

Try it.

ⓐ \(\sqrt[5]{64}\) ⓑ \(\sqrt[3]{256}\)

Try it.

ⓐ \(\sqrt[4]{3125}\) ⓑ \(\sqrt[3]{81}\)

Solution

ⓐ \(5\sqrt[4]{5}\) ⓑ \(3\sqrt[3]{3}\)

Try it.

ⓐ \(\sqrt[3]{108{x}^{5}}\) ⓑ \(\sqrt[4]{48{y}^{6}}\)

Try it.

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

Solution

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

Try it.

ⓐ \(\sqrt[4]{405{m}^{10}}\) ⓑ \(\sqrt[5]{160{n}^{8}}\)

Try it.

ⓐ \(\sqrt[3]{512{p}^{5}}\) ⓑ \(\sqrt[4]{324{q}^{7}}\)

Solution

ⓐ \(8p\sqrt[3]{{p}^{2}}\) ⓑ \(3\left|q\right|\sqrt[4]{4{q}^{3}}\)

Try it.

ⓐ \(\sqrt[3]{-864}\) ⓑ \(\sqrt[4]{-256}\)

Try it.

ⓐ \(\sqrt[5]{-486}\) ⓑ \(\sqrt[6]{-64}\)

Solution

ⓐ \(-3\sqrt[5]{2}\) ⓑ \(\text{not real}\)

Try it.

ⓐ \(\sqrt[5]{-32}\) ⓑ \(\sqrt[8]{-1}\)

Try it.

ⓐ \(\sqrt[3]{-8}\) ⓑ \(\sqrt[4]{-16}\)

Solution

ⓐ \(-2\) ⓑ \(\text{not real}\)

Use the Quotient Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

Try it.

ⓐ \(\sqrt[3]{\frac{{p}^{11}}{{p}^{2}}}\) ⓑ \(\sqrt[4]{\frac{{q}^{17}}{{q}^{13}}}\)

Try it.

ⓐ \(\sqrt[5]{\frac{{d}^{12}}{{d}^{7}}}\) ⓑ \(\sqrt[8]{\frac{{m}^{12}}{{m}^{4}}}\)

Solution

ⓐ \(d\) ⓑ \(|m|\)

Try it.

ⓐ \(\sqrt[5]{\frac{{u}^{21}}{{u}^{11}}}\) ⓑ \(\sqrt[6]{\frac{{v}^{30}}{{v}^{12}}}\)

Try it.

ⓐ \(\sqrt[3]{\frac{{r}^{14}}{{r}^{5}}}\) ⓑ \(\sqrt[4]{\frac{{c}^{21}}{{c}^{9}}}\)

Solution

ⓐ \({r}^{3}\) ⓑ \(|{c}^{3}|\)

Try it.

ⓐ \(\frac{\sqrt[4]{64}}{\sqrt[4]{2}}\) ⓑ \(\frac{\sqrt[5]{128{x}^{8}}}{\sqrt[5]{2{x}^{2}}}\)

Try it.

ⓐ \(\frac{\sqrt[3]{-625}}{\sqrt[3]{5}}\) ⓑ \(\frac{\sqrt[4]{80{m}^{7}}}{\sqrt[4]{5m}}\)

Solution

ⓐ \(-5\) ⓑ \(2\left|m\right|\sqrt[4]{{m}^{2}}\)

Try it.

ⓐ \(\sqrt[3]{\frac{1250}{2}}\) ⓑ \(\sqrt[4]{\frac{486{y}^{9}}{2{y}^{3}}}\)

Try it.

ⓐ \(\sqrt[3]{\frac{162}{6}}\) ⓑ \(\sqrt[4]{\frac{160{r}^{10}}{5{r}^{3}}}\)

Solution

ⓐ \(3\sqrt[3]{6}\) ⓑ \(2|r|\sqrt[4]{2{r}^{3}}\)

Try it.

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

Try it.

ⓐ \(\sqrt[5]{\frac{96{r}^{11}}{{s}^{3}}}\) ⓑ \(\sqrt[6]{\frac{128{u}^{7}}{{v}^{3}}}\)

Solution

ⓐ \(2{r}^{2}\sqrt[5]{\frac{3r}{{s}^{3}}}\) ⓑ \(2\left|u\right|\sqrt[6]{\frac{2u}{{v}^{3}}}\)

Try it.

ⓐ \(\sqrt[3]{\frac{81{s}^{8}}{{t}^{3}}}\) ⓑ \(\sqrt[4]{\frac{64{p}^{15}}{{q}^{12}}}\)

Try it.

ⓐ \(\sqrt[3]{\frac{625{u}^{10}}{{v}^{3}}}\) ⓑ \(\sqrt[4]{\frac{729{c}^{21}}{{d}^{8}}}\)

Solution

ⓐ \(\frac{5{u}^{3}\sqrt[3]{5u}}{v}\) ⓑ \(\frac{3\left|{c}^{5}\right|\sqrt[4]{9c}}{{d}^{2}}\)

Add and Subtract Higher Roots

In the following exercises, simplify.

Try it.

ⓐ \(\sqrt[7]{8p}+\sqrt[7]{8p}\) ⓑ \(3\sqrt[3]{25}-\sqrt[3]{25}\)

Try it.

ⓐ \(\sqrt[3]{15q}+\sqrt[3]{15q}\) ⓑ \(2\sqrt[4]{27}-6\sqrt[4]{27}\)

Solution

ⓐ \(2\sqrt[3]{15q}\) ⓑ \(-4\sqrt[4]{27}\)

Try it.

ⓐ \(3\sqrt[5]{9x}+7\sqrt[5]{9x}\) ⓑ \(8\sqrt[7]{3q}-2\sqrt[7]{3q}\)

Try it.

ⓐ ⓑ

Solution

ⓐ ⓑ

Try it.

ⓐ \(\sqrt[3]{81}-\sqrt[3]{192}\) ⓑ \(\sqrt[4]{512}-\sqrt[4]{32}\)

Try it.

ⓐ \(\sqrt[3]{250}-\sqrt[3]{54}\) ⓑ \(\sqrt[4]{243}-\sqrt[4]{1875}\)

Solution

ⓐ \(2\sqrt[3]{2}\) ⓑ \(-2\sqrt[4]{3}\)

Try it.

ⓐ \(\sqrt[3]{128}+\sqrt[3]{250}\) ⓑ \(\sqrt[5]{729}+\sqrt[5]{96}\)

Try it.

ⓐ \(\sqrt[4]{243}+\sqrt[4]{1250}\) ⓑ \(\sqrt[3]{2000}+\sqrt[3]{54}\)

Solution

ⓐ \(3\sqrt[4]{3}+5\sqrt[4]{2}\) ⓑ \(13\sqrt[3]{2}\)

Try it.

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

Try it.

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

Solution

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

Mixed Practice

In the following exercises, simplify.

Try it.

\(\sqrt[4]{16}\)

Try it.

\(\sqrt[6]{64}\)

Solution

\(2\)

Try it.

\(\sqrt[3]{{a}^{3}}\)

Try it.

Solution

\(|b|\)

Try it.

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

Try it.

\(\sqrt[3]{125{d}^{15}}\)

Solution

\(5{d}^{5}\)

Try it.

\(\sqrt[3]{{r}^{5}}\)

Try it.

\(\sqrt[4]{{s}^{10}}\)

Solution

\({s}^{2}\sqrt[4]{{s}^{2}}\)

Try it.

\(\sqrt[3]{108{x}^{5}}\)

Try it.

\(\sqrt[4]{48{y}^{6}}\)

Solution

\(2\left|y\right|\sqrt[4]{3{y}^{2}}\)

Try it.

\(\sqrt[5]{-486}\)

Try it.

\(\sqrt[6]{-64}\)

Solution

\(\text{not real}\)

Try it.

\(\frac{\sqrt[4]{64}}{\sqrt[4]{2}}\)

Try it.

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

Solution

\(2x\sqrt[5]{2x}\)

Try it.

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

Try it.

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

Solution

\(2\left|u\right|\sqrt[6]{\frac{2u}{{v}^{3}}}\)

Try it.

\(\sqrt[3]{81}-\sqrt[3]{192}\)

Try it.

\(\sqrt[4]{512}-\sqrt[4]{32}\)

Solution

\(2\sqrt[4]{2}\)

Try it.

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

Try it.

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

Solution

\(3│u│\sqrt[4]{6{u}^{3}}+4\sqrt[4]{3{u}^{3}}\)

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Simplify: \({y}^{5}{y}^{4}\).
If you missed this problem, review .

Odkrij odgovor

\({y}^{9}\)
Simplify: \({({n}^{2})}^{6}\).
If you missed this problem, review .

Odkrij odgovor

\({n}^{12}\)
Simplify: \(\frac{{x}^{8}}{{x}^{3}}\).
If you missed this problem, review .

Odkrij odgovor

\({x}^{5}\)
Simplify: ⓐ \(\sqrt[3]{8}\) ⓑ \(\sqrt[4]{81}\) ⓒ \(\sqrt[5]{32}\).

Odkrij odgovor

ⓐ

\(\sqrt[3]{8}\)
Since \({(2)}^{3}=8.\) \(2\)
ⓑ

\(\sqrt[4]{81}\)
Since \({(3)}^{4}=81.\) \(3\)
ⓒ

\(\sqrt[5]{32}\)
Since \({(2)}^{5}=32.\) \(2\)
Simplify: ⓐ \(\sqrt[3]{-64}\) ⓑ \(\sqrt[4]{-16}\) ⓒ \(\sqrt[5]{-243}\).

Odkrij odgovor

ⓐ

\(\sqrt[3]{-64}\)
Since \({(-4)}^{3}=-64.\) \(-4\)
ⓑ

\(\sqrt[4]{-16}\)
Think, \({(?)}^{4}=-16.\) No real number raised to the fourth power is negative. Not a real number.

ⓒ

\(\sqrt[5]{-243}\)
Since \({(-3)}^{5}=-243.\) \(-3\)
Simplify: ⓐ \(\sqrt{{x}^{2}}\) ⓑ \(\sqrt[3]{{n}^{3}}\) ⓒ \(\sqrt[4]{{p}^{4}}\) ⓓ \(\sqrt[5]{{y}^{5}}\).

Odkrij odgovor

We use the absolute value to be sure to get the positive root.
ⓐ

\(\sqrt{{x}^{2}}\)
Since \({(x)}^{2}={x}^{2}\) and we want the positive root. \(|x|\)

ⓑ

\(\sqrt[3]{{n}^{3}}\)
Since \({(n)}^{3}={n}^{3}.\) It is an odd root so there is no need for an absolute value sign. \(n\)
ⓒ

\(\sqrt[4]{{p}^{4}}\)
Since \({(p)}^{4}={p}^{4}\) and we want the positive root. \(|p|\)

ⓓ

\(\sqrt[5]{{y}^{5}}\)
Since \({(y)}^{5}={y}^{5}.\) It is an odd root so there is no need for an absolute value sign. \(y\)
Simplify: ⓐ \(\sqrt{{b}^{2}}\) ⓑ \(\sqrt[3]{{w}^{3}}\) ⓒ \(\sqrt[4]{{m}^{4}}\) ⓓ \(\sqrt[5]{{q}^{5}}\).

Odkrij odgovor

ⓐ \(|b|\) ⓑ \(w\) ⓒ \(|m|\) ⓓ \(q\)
Simplify: ⓐ \(\sqrt{{y}^{2}}\) ⓑ \(\sqrt[3]{{p}^{3}}\) ⓒ \(\sqrt[4]{{z}^{4}}\) ⓓ \(\sqrt[5]{{q}^{5}}\).

Odkrij odgovor

ⓐ \(|y|\) ⓑ \(p\) ⓒ \(|z|\) ⓓ \(q\)
Simplify: ⓐ \(\sqrt[3]{{y}^{18}}\) ⓑ \(\sqrt[4]{{z}^{8}}\).

Odkrij odgovor

ⓐ

\(\sqrt[3]{{y}^{18}}\)
Since \({({y}^{6})}^{3}={y}^{18}.\) \(\sqrt[3]{{({y}^{6})}^{3}}\)
\({y}^{6}\)
ⓑ

\(\sqrt[4]{{z}^{8}}\)
Since \({({z}^{2})}^{4}={z}^{8}.\) \(\sqrt[4]{{({z}^{2})}^{4}}\)
Since \({z}^{2}\) is positive, we do not need an absolute value sign. \({z}^{2}\)
Simplify: ⓐ \(\sqrt[4]{{u}^{12}}\) ⓑ \(\sqrt[3]{{v}^{15}}\).

Odkrij odgovor

ⓐ \(\left|{u}^{3}\right|\) ⓑ \({v}^{5}\)
Simplify: ⓐ \(\sqrt[5]{{c}^{20}}\) ⓑ \(\sqrt[6]{{d}^{24}}\).

Odkrij odgovor

ⓐ \({c}^{4}\) ⓑ \({d}^{4}\)
Simplify: ⓐ \(\sqrt[3]{64{p}^{6}}\) ⓑ \(\sqrt[4]{16{q}^{12}}\).

Odkrij odgovor

ⓐ

\(\sqrt[3]{64{p}^{6}}\)
Rewrite \(64{p}^{6}\ \text{as}\ {(4{p}^{2})}^{3}.\) \(\sqrt[3]{{(4{p}^{2})}^{3}}\)
Take the cube root. \(4{p}^{2}\)
ⓑ

\(\sqrt[4]{16{q}^{12}}\)
Rewrite the radicand as a fourth power. \(\sqrt[4]{{(2{q}^{3})}^{4}}\)
Take the fourth root. \(2|{q}^{3}|\)
Simplify: ⓐ \(\sqrt[3]{27{x}^{27}}\) ⓑ \(\sqrt[4]{81{q}^{28}}\).

Odkrij odgovor

ⓐ \(3{x}^{9}\) ⓑ \(3|{q}^{7}|\)
Simplify: ⓐ \(\sqrt[3]{125{p}^{9}}\) ⓑ \(\sqrt[5]{243{q}^{25}}\).

Odkrij odgovor

ⓐ \(5{p}^{3}\) ⓑ \(3{q}^{5}\)

Simplify: ⓐ \(\sqrt[3]{{x}^{4}}\) ⓑ \(\sqrt[4]{{x}^{7}}\).

Odkrij odgovor

ⓐ

	\(\sqrt[3]{{x}^{4}}\)
Rewrite the radicand as a product using the largest perfect cube factor.	\(\sqrt[3]{{x}^{3}\cdot x}\)
Rewrite the radical as the product of two radicals.	\(\sqrt[3]{{x}^{3}}\cdot \sqrt[3]{x}\)
Simplify.	\(x\sqrt[3]{x}\)

ⓑ

	\(\sqrt[4]{{x}^{7}}\)
Rewrite the radicand as a product using the greatest perfect fourth power factor.	\(\sqrt[4]{{x}^{4}\cdot {x}^{3}}\)
Rewrite the radical as the product of two radicals.	\(\sqrt[4]{{x}^{4}}\cdot \sqrt[4]{{x}^{3}}\)
Simplify.	\(\|x\|\sqrt[4]{{x}^{3}}\)

Simplify: ⓐ \(\sqrt[4]{{y}^{6}}\) ⓑ \(\sqrt[3]{{z}^{5}}\).

Odkrij odgovor

ⓐ \(|y|\sqrt[4]{{y}^{2}}\) ⓑ \(z\sqrt[3]{{z}^{2}}\)
Simplify: ⓐ \(\sqrt[5]{{p}^{8}}\) ⓑ \(\sqrt[6]{{q}^{13}}\).

Odkrij odgovor

ⓐ \(p\sqrt[5]{{p}^{3}}\) ⓑ \({q}^{2}\sqrt[6]{q}\)

Simplify: ⓐ \(\sqrt[3]{16}\) ⓑ \(\sqrt[4]{243}\).

Odkrij odgovor

ⓐ

	\(\sqrt[3]{16}\)
	\(\sqrt[3]{{2}^{4}}\)
Rewrite the radicand as a product using the greatest perfect cube factor.	\(\sqrt[3]{{2}^{3}\cdot 2}\)
Rewrite the radical as the product of two radicals.	\(\sqrt[3]{{2}^{3}}\cdot \sqrt[3]{2}\)
Simplify.	\(2\sqrt[3]{2}\)

ⓑ

	\(\sqrt[4]{243}\)
	\(\sqrt[4]{{3}^{5}}\)
Rewrite the radicand as a product using the greatest perfect fourth power factor.	\(\sqrt[4]{{3}^{4}\cdot 3}\)
Rewrite the radical as the product of two radicals.	\(\sqrt[4]{{3}^{4}}\cdot \sqrt[4]{3}\)
Simplify.	\(3\sqrt[4]{3}\)

Simplify: ⓐ \(\sqrt[3]{81}\) ⓑ \(\sqrt[4]{64}\).

Odkrij odgovor

ⓐ \(3\sqrt[3]{3}\) ⓑ \(2\sqrt[4]{4}\)
Simplify: ⓐ \(\sqrt[3]{625}\) ⓑ \(\sqrt[4]{729}\).

Odkrij odgovor

ⓐ \(5\sqrt[3]{5}\) ⓑ \(3\sqrt[4]{9}\)

Simplify: ⓐ \(\sqrt[3]{24{x}^{7}}\) ⓑ \(\sqrt[4]{80{y}^{14}}\).

Odkrij odgovor

ⓐ

	\(\sqrt[3]{24{x}^{7}}\)
Rewrite the radicand as a product using perfect cube factors.	\(\sqrt[3]{{2}^{3}{x}^{6}\cdot 3x}\)
Rewrite the radical as the product of two radicals.	\(\sqrt[3]{{2}^{3}{x}^{6}}\cdot \sqrt[3]{3x}\)
Rewrite the first radicand as \({(2{x}^{2})}^{3}.\)	\(\sqrt[3]{{(2{x}^{2})}^{3}}\cdot \sqrt[3]{3x}\)
Simplify.	\(2{x}^{2}\sqrt[3]{3x}\)

ⓑ

	\(\sqrt[4]{80{y}^{14}}\)
Rewrite the radicand as a product using perfect fourth power factors.	\(\sqrt[4]{{2}^{4}{y}^{12}\cdot 5{y}^{2}}\)
Rewrite the radical as the product of two radicals.	\(\sqrt[4]{{2}^{4}{y}^{12}}\cdot \sqrt[4]{5{y}^{2}}\)
Rewrite the first radicand as \({(2{y}^{3})}^{4}.\)	\(\sqrt[4]{{(2{y}^{3})}^{4}}\cdot \sqrt[4]{5{y}^{2}}\)
Simplify.	\(2\|{y}^{3}\|\sqrt[4]{5{y}^{2}}\)

Simplify: ⓐ \(\sqrt[3]{54{p}^{10}}\) ⓑ \(\sqrt[4]{64{q}^{10}}\).

Odkrij odgovor

ⓐ \(3{p}^{3}\sqrt[3]{2p}\) ⓑ \(2{q}^{2}\sqrt[4]{4{q}^{2}}\)
Simplify: ⓐ \(\sqrt[3]{128{m}^{11}}\) ⓑ \(\sqrt[4]{162{n}^{7}}\).

Odkrij odgovor

ⓐ \(4{m}^{3}\sqrt[3]{2{m}^{2}}\) ⓑ \(3|n|\sqrt[4]{2{n}^{3}}\)
Simplify: ⓐ \(\sqrt[3]{-27}\) ⓑ \(\sqrt[4]{-16}\).

Odkrij odgovor

ⓐ

\(\sqrt[3]{-27}\)
Rewrite the radicand as a product using perfect cube factors. \(\sqrt[3]{{(-3)}^{3}}\)
Take the cube root. \(-3\)

ⓑ

\(\sqrt[4]{-16}\)
There is no real number \(n\) where \({n}^{4}=-16.\) Not a real number.
Simplify: ⓐ \(\sqrt[3]{-108}\) ⓑ \(\sqrt[4]{-48}\).

Odkrij odgovor

ⓐ \(-3\sqrt[3]{4}\) ⓑ \(\text{not real}\)
Simplify: ⓐ \(\sqrt[3]{-625}\) ⓑ \(\sqrt[4]{-324}\).

Odkrij odgovor

ⓐ \(-5\sqrt[3]{5}\) ⓑ \(\text{not real}\)

Simplify: ⓐ \(\sqrt[3]{\frac{{a}^{8}}{{a}^{5}}}\) ⓑ \(\sqrt[4]{\frac{{a}^{10}}{{a}^{2}}}\).

Odkrij odgovor

ⓐ

	\(\sqrt[3]{\frac{{a}^{8}}{{a}^{5}}}\)
Simplify the fraction under the radical first.	\(\sqrt[3]{{a}^{3}}\)
Simplify.	\(a\)

ⓑ

	\(\sqrt[4]{\frac{{a}^{10}}{{a}^{2}}}\)
Simplify the fraction under the radical first.	\(\sqrt[4]{{a}^{8}}\)
Rewrite the radicand using perfect fourth power factors.	\(\sqrt[4]{{({a}^{2})}^{4}}\)
Simplify.	\({a}^{2}\)

Simplify: ⓐ \(\sqrt[4]{\frac{{x}^{7}}{{x}^{3}}}\) ⓑ \(\sqrt[4]{\frac{{y}^{17}}{{y}^{5}}}\).

Odkrij odgovor

ⓐ \(|x|\) ⓑ \(\left|{y}^{3}\right|\)
Simplify: ⓐ \(\sqrt[3]{\frac{{m}^{13}}{{m}^{7}}}\) ⓑ \(\sqrt[5]{\frac{{n}^{12}}{{n}^{2}}}\).

Odkrij odgovor

ⓐ \({m}^{2}\) ⓑ \({n}^{2}\)

Simplify: ⓐ \(\frac{\sqrt[3]{-108}}{\sqrt[3]{2}}\) ⓑ \(\frac{\sqrt[4]{96{x}^{7}}}{\sqrt[4]{3{x}^{2}}}\).

Odkrij odgovor

ⓐ

	\(\frac{\sqrt[3]{-108}}{\sqrt[3]{2}}\)
Neither radicand is a perfect cube, so use the Quotient Property to write as one radical.	\(\sqrt[3]{\frac{-108}{2}}\)
Simplify the fraction under the radical.	\(\sqrt[3]{-54}\)
Rewrite the radicand as a product using perfect cube factors.	\(\sqrt[3]{{(-3)}^{3}\cdot 2}\)
Rewrite the radical as the product of two radicals.	\(\sqrt[3]{{(-3)}^{3}}\cdot \sqrt[3]{2}\)
Simplify.	\(-3\sqrt[3]{2}\)

ⓑ

	\(\frac{\sqrt[4]{96{x}^{7}}}{\sqrt[4]{3{x}^{2}}}\)
Neither radicand is a perfect fourth power, so use the Quotient Property to write as one radical.	\(\sqrt[4]{\frac{96{x}^{7}}{3{x}^{2}}}\)
Simplify the fraction under the radical.	\(\sqrt[4]{32{x}^{5}}\)
Rewrite the radicand as a product using perfect fourth power factors.	\(\sqrt[4]{{2}^{4}{x}^{4}\cdot 2x}\)
Rewrite the radical as the product of two radicals.	\(\sqrt[4]{{(2x)}^{4}}\cdot \sqrt[4]{2x}\)
Simplify.	\(2\|x\|\sqrt[4]{2x}\)

Simplify: ⓐ \(\frac{\sqrt[3]{-532}}{\sqrt[3]{2}}\) ⓑ \(\frac{\sqrt[4]{486{m}^{11}}}{\sqrt[4]{3{m}^{5}}}\).

Odkrij odgovor

ⓐ \(-\sqrt[3]{266}\) ⓑ \(3|m|\sqrt[4]{2{m}^{2}}\)
Simplify: ⓐ \(\frac{\sqrt[3]{-192}}{\sqrt[3]{3}}\) ⓑ \(\frac{\sqrt[4]{324{n}^{7}}}{\sqrt[4]{2{n}^{3}}}\).

Odkrij odgovor

ⓐ \(-4\) ⓑ \(3|n|\sqrt[4]{2}\)

Simplify: ⓐ \(\sqrt[3]{\frac{24{x}^{7}}{{y}^{3}}}\) ⓑ \(\sqrt[4]{\frac{48{x}^{10}}{{y}^{8}}}\).

Odkrij odgovor

ⓐ

	\(\sqrt[3]{\frac{24{x}^{7}}{{y}^{3}}}\)
The fraction in the radicand cannot be simplified. Use the Quotient Property to write as two radicals.	\(\frac{\sqrt[3]{24{x}^{7}}}{\sqrt[3]{{y}^{3}}}\)
Rewrite each radicand as a product using perfect cube factors.	\(\frac{\sqrt[3]{8{x}^{6}\cdot 3x}}{\sqrt[3]{{y}^{3}}}\)
Rewrite the numerator as the product of two radicals.	\(\frac{\sqrt[3]{{(2{x}^{2})}^{3}}\ \sqrt[3]{3x}}{\sqrt[3]{{y}^{3}}}\)
Simplify.	\(\frac{2{x}^{2}\sqrt[3]{3x}}{y}\)

ⓑ

	\(\sqrt[4]{\frac{48{x}^{10}}{{y}^{8}}}\)
The fraction in the radicand cannot be simplified. Use the Quotient Property to write as two radicals.	\(\frac{\sqrt[4]{48{x}^{10}}}{\sqrt[4]{{y}^{8}}}\)
Rewrite each radicand as a product using perfect fourth power factors.	\(\frac{\sqrt[4]{16{x}^{8}\cdot 3{x}^{2}}}{\sqrt[4]{{y}^{8}}}\)
Rewrite the numerator as the product of two radicals.	\(\frac{\sqrt[4]{{(2{x}^{2})}^{4}}\ \sqrt[4]{3{x}^{2}}}{\sqrt[4]{{({y}^{2})}^{4}}}\)
Simplify.	\(\frac{2{x}^{2}\sqrt[4]{3{x}^{2}}}{{y}^{2}}\)

Simplify: ⓐ \(\sqrt[3]{\frac{108{c}^{10}}{{d}^{6}}}\) ⓑ \(\sqrt[4]{\frac{80{x}^{10}}{{y}^{5}}}\).

Odkrij odgovor

ⓐ \(\frac{3{c}^{3}\sqrt[3]{4c}}{{d}^{2}}\) ⓑ \(\frac{2{x}^{2}}{|y|}\ \sqrt[4]{\frac{5{x}^{2}}{y}}\)
Simplify: ⓐ \(\sqrt[3]{\frac{40{r}^{3}}{s}}\) ⓑ \(\sqrt[4]{\frac{162{m}^{14}}{{n}^{12}}}\).

Odkrij odgovor

ⓐ \(2r\sqrt[3]{\frac{5}{s}}\) ⓑ \(\frac{3\left|{m}^{3}\right|\sqrt[4]{2{m}^{2}}}{|{n}^{3}|}\)
Simplify: ⓐ \(\sqrt[3]{4x}+\sqrt[3]{4x}\) ⓑ \(4\sqrt[4]{8}-2\sqrt[4]{8}\).

Odkrij odgovor

ⓐ

\(\sqrt[3]{4x}+\sqrt[3]{4x}\)
The radicals are like, so we add the coefficients. \(2\sqrt[3]{4x}\)

ⓑ

\(4\sqrt[4]{8}-2\sqrt[4]{8}\)
The radicals are like, so we subtract the coefficients. \(2\sqrt[4]{8}\)

Symbols used here

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\neq

not equal
The two sides are different.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

imaginary unit
i² = −1.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Higher Roots

Simplify expressions with higher roots
Use the Product Property to simplify expressions with higher roots
Use the Quotient Property to simplify expressions with higher roots
Add and subtract higher roots
When
For any integer
For any integer
To combine like radicals, simply add or subtract the coefficients while keeping the radical the same.

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

Poskusi sam.

Parts of this page are adapted from OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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	\(\sqrt{{x}^{2}}\)
Since \({(x)}^{2}={x}^{2}\) and we want the positive root.	\(\|x\|\)

	\(\sqrt[3]{{n}^{3}}\)
Since \({(n)}^{3}={n}^{3}.\) It is an odd root so there is no need for an absolute value sign.	\(n\)

	\(\sqrt[4]{{p}^{4}}\)
Since \({(p)}^{4}={p}^{4}\) and we want the positive root.	\(\|p\|\)

	\(\sqrt[5]{{y}^{5}}\)
Since \({(y)}^{5}={y}^{5}.\) It is an odd root so there is no need for an absolute value sign.	\(y\)

	\(\sqrt[3]{{y}^{18}}\)
Since \({({y}^{6})}^{3}={y}^{18}.\)	\(\sqrt[3]{{({y}^{6})}^{3}}\)
	\({y}^{6}\)

	\(\sqrt[4]{{z}^{8}}\)
Since \({({z}^{2})}^{4}={z}^{8}.\)	\(\sqrt[4]{{({z}^{2})}^{4}}\)
Since \({z}^{2}\) is positive, we do not need an absolute value sign.	\({z}^{2}\)

	\(\sqrt[3]{64{p}^{6}}\)
Rewrite \(64{p}^{6}\ \text{as}\ {(4{p}^{2})}^{3}.\)	\(\sqrt[3]{{(4{p}^{2})}^{3}}\)
Take the cube root.	\(4{p}^{2}\)

	\(\sqrt[4]{16{q}^{12}}\)
Rewrite the radicand as a fourth power.	\(\sqrt[4]{{(2{q}^{3})}^{4}}\)
Take the fourth root.	\(2\|{q}^{3}\|\)

	\(\sqrt[3]{-27}\)
Rewrite the radicand as a product using perfect cube factors.	\(\sqrt[3]{{(-3)}^{3}}\)
Take the cube root.	\(-3\)

	\(\sqrt[4]{-16}\)
There is no real number \(n\) where \({n}^{4}=-16.\)	Not a real number.