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Simplify Radical Expressions

Use the Product Property to simplify radical expressions

Use the Product Property to Simplify Radical Expressions

We will simplify radical expressions in a way similar to how we simplified fractions. A fraction is simplified if there are no common factors in the numerator and denominator. To simplify a fraction, we look for any common factors in the numerator and denominator.

A radical expression, \(\sqrt[n]{a},\) is considered simplified if it has no factors of \({m}^{n}.\) So, to simplify a radical expression, we look for any factors in the radicand that are powers of the index.

For example, \(\sqrt{5}\) is considered simplified because there are no perfect square factors in 5. But \(\sqrt{12}\) is not simplified because 12 has a perfect square factor of 4.

Similarly, \(\sqrt[3]{4}\) is simplified because there are no perfect cube factors in 4. But \(\sqrt[3]{24}\) is not simplified because 24 has a perfect cube factor of 8.

To simplify radical expressions, we will also use some properties of roots. The properties we will use to simplify radical expressions are similar to the properties of exponents. We know that \({(ab)}^{n}={a}^{n}{b}^{n}.\) The corresponding of Product Property of Roots says that \(\sqrt[n]{ab}=\sqrt[n]{a}\cdot \sqrt[n]{b}.\)

We use the Product Property of Roots to remove all perfect square factors from a square root.

Simplify Square Roots Using the Product Property of Roots

Try it.

Simplify: \(\sqrt{98}.\)

Solution

Notice in the previous example that the simplified form of \(\sqrt{98}\) is \(7\sqrt{2},\) which is the product of an integer and a square root. We always write the integer in front of the square root.

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

Use the Quotient Property to Simplify Radical Expressions

Whenever you have to simplify a radical expression, the first step you should take is to determine whether the radicand is a perfect power of the index. If not, check the numerator and denominator for any common factors, and remove them. You may find a fraction in which both the numerator and the denominator are perfect powers of the index.

Example

Try it.

Simplify: ⓐ \(\sqrt{\frac{45}{80}}\) ⓑ \(\sqrt[3]{\frac{16}{54}}\) ⓒ \(\sqrt[4]{\frac{5}{80}}.\)

Solution

ⓐ

	\(\sqrt{\frac{45}{80}}\)
Simplify inside the radical first.
Rewrite showing the common factors of the numerator and denominator.	\(\sqrt{\frac{5\cdot 9}{5\cdot 16}}\)
Simplify the fraction by removing common factors.	\(\sqrt{\frac{9}{16}}\)
Simplify. Note \({(\frac{3}{4})}^{2}=\frac{9}{16}.\)	\(\frac{3}{4}\)

ⓑ

	\(\sqrt[3]{\frac{16}{54}}\)
Simplify inside the radical first.
Rewrite showing the common factors of the numerator and denominator.	\(\sqrt[3]{\frac{2\cdot 8}{2\cdot 27}}\)
Simplify the fraction by removing common factors.	\(\sqrt[3]{\frac{8}{27}}\)
Simplify. Note \({(\frac{2}{3})}^{3}=\frac{8}{27}.\)	\(\frac{2}{3}\)

ⓒ

	\(\sqrt[4]{\frac{5}{80}}\)
Simplify inside the radical first.
Rewrite showing the common factors of the numerator and denominator.	\(\sqrt[4]{\frac{5\cdot 1}{5\cdot 16}}\)
Simplify the fraction by removing common factors.	\(\sqrt[4]{\frac{1}{16}}\)
Simplify. Note \({(\frac{1}{2})}^{4}=\frac{1}{16}.\)	\(\frac{1}{2}\)

In the last example, our first step was to simplify the fraction under the radical by removing common factors. In the next example we will use the Quotient Property to simplify under the radical. We divide the like bases by subtracting their exponents,

\[\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},\ a\ne 0\]

Example

Try it.

Simplify: ⓐ \(\sqrt{\frac{{m}^{6}}{{m}^{4}}}\) ⓑ \(\sqrt[3]{\frac{{a}^{8}}{{a}^{5}}}\) ⓒ \(\sqrt[4]{\frac{{a}^{10}}{{a}^{2}}}.\)

Solution

ⓐ

	\(\sqrt{\frac{{m}^{6}}{{m}^{4}}}\)
Simplify the fraction inside the radical first.
Divide the like bases by subtracting the exponents.	\(\sqrt{{m}^{2}}\)
Simplify.	\(\|m\|\)

ⓑ

	\(\sqrt[3]{\frac{{a}^{8}}{{a}^{5}}}\)
Use the Quotient Property of exponents to simplify the fraction under the radical first.	\(\sqrt[3]{{a}^{3}}\)
Simplify.	\(a\)

ⓒ

	\(\sqrt[4]{\frac{{a}^{10}}{{a}^{2}}}\)
Use the Quotient Property of exponents to 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.	\({a}^{2}\)

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

Key Concepts

Simplified Radical Expression
- For real numbers a, m and \(n\ge 2\)
  \(\sqrt[n]{a}\) is considered simplified if a has no factors of \({m}^{n}\)
Product Property of n^th Roots
- For any real numbers, \(\sqrt[n]{a}\) and \(\sqrt[n]{b},\) and for any integer \(n\ge 2\)
  \(\sqrt[n]{ab}=\sqrt[n]{a}\cdot \sqrt[n]{b}\) and \(\sqrt[n]{a}\cdot \sqrt[n]{b}=\sqrt[n]{ab}\)
How to simplify a radical expression using the Product Property
1. Find the largest factor in the radicand that is a perfect power of the index.
  Rewrite the radicand as a product of two factors, using that factor.
2. Use the product rule to rewrite the radical as the product of two radicals.
3. Simplify the root of the perfect power.
Quotient Property of Radical Expressions
- If \(\sqrt[n]{a}\) and \(\sqrt[n]{b}\) are real numbers, \(b\ne 0,\) and for any integer \(n\ge 2\) then,
  \(\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}\) and \(\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}\)
How to simplify a radical expression using the Quotient Property.
1. Simplify the fraction in the radicand, if possible.
2. Use the Quotient Property to rewrite the radical as the quotient of two radicals.
3. Simplify the radicals in the numerator and the denominator.

Simplify Radical Expressions

Use the Product Property to Simplify Radical Expressions

In the following exercises, use the Product Property to simplify radical expressions.

Try it.

\(\sqrt{27}\)

Solution

\(3\sqrt{3}\)

Try it.

\(\sqrt{80}\)

Try it.

\(\sqrt{125}\)

Solution

\(5\sqrt{5}\)

Try it.

\(\sqrt{96}\)

Try it.

\(\sqrt{147}\)

Solution

\(7\sqrt{3}\)

Try it.

\(\sqrt{450}\)

Try it.

\(\sqrt{800}\)

Solution

\(20\sqrt{2}\)

Try it.

\(\sqrt{675}\)

Try it.

ⓐ \(\sqrt[4]{32}\) ⓑ \(\sqrt[5]{64}\)

Solution

ⓐ \(2\sqrt[4]{2}\) ⓑ \(2\sqrt[5]{2}\)

Try it.

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

Try it.

ⓐ \(\sqrt[4]{64}\) ⓑ \(\sqrt[3]{256}\)

Solution

ⓐ \(2\sqrt[4]{4}\) ⓑ \(4\sqrt[3]{4}\)

Try it.

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

In the following exercises, simplify using absolute value signs as needed.

Try it.

ⓐ \(\sqrt{{y}^{11}}\) ⓑ \(\sqrt[3]{{r}^{5}}\) ⓒ \(\sqrt[4]{{s}^{10}}\)

Solution

ⓐ \(|{y}^{5}|\sqrt[]{y}\) ⓑ \(r\sqrt[3]{{r}^{2}}\) ⓒ \({s}^{2}\sqrt[4]{{s}^{2}}\)

Try it.

ⓐ \(\sqrt{{m}^{13}}\) ⓑ \(\sqrt[5]{{u}^{7}}\) ⓒ \(\sqrt[6]{{v}^{11}}\)

Try it.

ⓐ \(\sqrt{{n}^{21}}\) ⓑ \(\sqrt[3]{{q}^{8}}\) ⓒ \(\sqrt[8]{{n}^{10}}\)

Solution

ⓐ \({n}^{10}\sqrt{n}\) ⓑ \({q}^{2}\sqrt[3]{{q}^{2}}\)
ⓒ \(|n|\sqrt[8]{{n}^{2}}\)

Try it.

ⓐ \(\sqrt{{r}^{25}}\) ⓑ \(\sqrt[5]{{p}^{8}}\) ⓒ \(\sqrt[4]{{m}^{5}}\)

Try it.

ⓐ \(\sqrt{125{r}^{13}}\) ⓑ \(\sqrt[3]{108{x}^{5}}\) ⓒ \(\sqrt[4]{48{y}^{6}}\)

Solution

ⓐ \(5{r}^{6}\sqrt[]{5r}\) ⓑ \(3x\sqrt[3]{4{x}^{2}}\)
ⓒ \(2|y|\sqrt[4]{3{y}^{2}}\)

Try it.

ⓐ \(\sqrt{80{s}^{15}}\) ⓑ \(\sqrt[5]{96{a}^{7}}\) ⓒ \(\sqrt[6]{128{b}^{7}}\)

Try it.

ⓐ \(\sqrt{242{m}^{23}}\) ⓑ \(\sqrt[4]{405{m}^{10}}\) ⓒ \(\sqrt[5]{160{n}^{8}}\)

Solution

ⓐ \(11|{m}^{11}|\sqrt[]{2m}\) ⓑ \(3{m}^{2}\sqrt[4]{5{m}^{2}}\) ⓒ \(2n\sqrt[5]{5{n}^{3}}\)

Try it.

ⓐ \(\sqrt{175{n}^{13}}\) ⓑ \(\sqrt[5]{512{p}^{5}}\) ⓒ \(\sqrt[4]{324{q}^{7}}\)

Try it.

ⓐ \(\sqrt{147{m}^{7}{n}^{11}}\) ⓑ \(\sqrt[3]{48{x}^{6}{y}^{7}}\) ⓒ \(\sqrt[4]{32{x}^{5}{y}^{4}}\)

Solution

ⓐ \(7|{m}^{3}{n}^{5}|\sqrt[]{3mn}\) ⓑ \(2{x}^{2}{y}^{2}\sqrt[3]{6y}\) ⓒ \(2|xy|\sqrt[4]{2x}\)

Try it.

ⓐ \(\sqrt{96{r}^{3}{s}^{3}}\) ⓑ \(\sqrt[3]{80{x}^{7}{y}^{6}}\) ⓒ \(\sqrt[4]{80{x}^{8}{y}^{9}}\)

Try it.

ⓐ \(\sqrt{192{q}^{3}{r}^{7}}\) ⓑ \(\sqrt[3]{54{m}^{9}{n}^{10}}\) ⓒ \(\sqrt[4]{81{a}^{9}{b}^{8}}\)

Solution

ⓐ \(8|q{r}^{3}|\sqrt{3qr}\) ⓑ \(3{m}^{3}{n}^{3}\sqrt[3]{2n}\) ⓒ \(3{a}^{2}{b}^{2}\sqrt[4]{a}\)

Try it.

ⓐ \(\sqrt{150{m}^{9}{n}^{3}}\) ⓑ \(\sqrt[3]{81{p}^{7}{q}^{8}}\) ⓒ \(\sqrt[4]{162{c}^{11}{d}^{12}}\)

Try it.

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

Solution

ⓐ \(-6\sqrt[3]{4}\) ⓑ not real

Try it.

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

Try it.

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

Solution

ⓐ \(-2\) ⓑ not real

Try it.

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

Try it.

ⓐ \(5+\sqrt{12}\) ⓑ \(\frac{10-\sqrt{24}}{2}\)

Solution

ⓐ \(5+2\sqrt{3}\) ⓑ \(5-\sqrt{6}\)

Try it.

ⓐ \(8+\sqrt{96}\) ⓑ \(\frac{8-\sqrt{80}}{4}\)

Try it.

ⓐ \(1+\sqrt{45}\) ⓑ \(\frac{3+\sqrt{90}}{3}\)

Solution

ⓐ \(1+3\sqrt{5}\) ⓑ \(1+\sqrt{10}\)

Try it.

ⓐ \(3+\sqrt{125}\) ⓑ \(\frac{15+\sqrt{75}}{5}\)

Use the Quotient Property to Simplify Radical Expressions

In the following exercises, use the Quotient Property to simplify square roots.

Try it.

ⓐ \(\sqrt{\frac{45}{80}}\) ⓑ \(\sqrt[3]{\frac{8}{27}}\) ⓒ \(\sqrt[4]{\frac{1}{81}}\)

Solution

ⓐ \(\frac{3}{4}\) ⓑ \(\frac{2}{3}\) ⓒ \(\frac{1}{3}\)

Try it.

ⓐ \(\sqrt{\frac{72}{98}}\) ⓑ \(\sqrt[3]{\frac{24}{81}}\) ⓒ \(\sqrt[4]{\frac{6}{96}}\)

Try it.

ⓐ \(\sqrt{\frac{100}{36}}\) ⓑ \(\sqrt[3]{\frac{81}{375}}\) ⓒ \(\sqrt[4]{\frac{1}{256}}\)

Solution

ⓐ \(\frac{5}{3}\) ⓑ \(\frac{3}{5}\) ⓒ \(\frac{1}{4}\)

Try it.

ⓐ \(\sqrt{\frac{121}{16}}\) ⓑ \(\sqrt[3]{\frac{16}{250}}\) ⓒ \(\sqrt[4]{\frac{32}{162}}\)

Try it.

ⓐ \(\sqrt{\frac{{x}^{10}}{{x}^{6}}}\) ⓑ \(\sqrt[3]{\frac{{p}^{11}}{{p}^{2}}}\) ⓒ \(\sqrt[4]{\frac{{q}^{17}}{{q}^{13}}}\)

Solution

ⓐ \({x}^{2}\) ⓑ \({p}^{3}\) ⓒ \(|q|\)

Try it.

ⓐ \(\sqrt{\frac{{p}^{20}}{{p}^{10}}}\) ⓑ \(\sqrt[5]{\frac{{d}^{12}}{{d}^{7}}}\) ⓒ \(\sqrt[8]{\frac{{m}^{12}}{{m}^{4}}}\)

Try it.

ⓐ \(\sqrt{\frac{{y}^{4}}{{y}^{8}}}\) ⓑ \(\sqrt[5]{\frac{{u}^{21}}{{u}^{11}}}\) ⓒ \(\sqrt[6]{\frac{{v}^{30}}{{v}^{12}}}\)

Solution

ⓐ \(\frac{1}{{y}^{2}}\) ⓑ \({u}^{2}\) ⓒ \(|{v}^{3}|\)

Try it.

ⓐ \(\sqrt{\frac{{q}^{8}}{{q}^{14}}}\) ⓑ \(\sqrt[3]{\frac{{r}^{14}}{{r}^{5}}}\) ⓒ \(\sqrt[4]{\frac{{c}^{21}}{{c}^{9}}}\)

Try it.

\(\sqrt{\frac{96{x}^{7}}{121}}\)

Solution

\(\frac{4|{x}^{3}|\sqrt{6x}}{11}\)

Try it.

\(\sqrt{\frac{108{y}^{4}}{49}}\)

Try it.

\(\sqrt{\frac{300{m}^{5}}{64}}\)

Solution

\(\frac{5{m}^{2}\sqrt{3m}}{4}\)

Try it.

\(\sqrt{\frac{125{n}^{7}}{169}}\)

Try it.

\(\sqrt{\frac{98{r}^{5}}{100}}\)

Solution

\(\frac{7{r}^{2}\sqrt{2r}}{10}\)

Try it.

\(\sqrt{\frac{180{s}^{10}}{144}}\)

Try it.

\(\sqrt{\frac{28{q}^{6}}{225}}\)

Solution

\(\frac{2|{q}^{3}|\sqrt{7}}{15}\)

Try it.

\(\sqrt{\frac{150{r}^{3}}{256}}\)

Try it.

ⓐ \(\sqrt{\frac{75{r}^{9}}{{s}^{8}}}\) ⓑ \(\sqrt[3]{\frac{54{a}^{8}}{{b}^{3}}}\) ⓒ \(\sqrt[4]{\frac{64{c}^{5}}{{d}^{4}}}\)

Solution

ⓐ \(\frac{5{r}^{4}\sqrt{3r}}{{s}^{4}}\) ⓑ \(\frac{3{a}^{2}\sqrt[3]{2{a}^{2}}}{b}\)
ⓒ \(\frac{2|c|\sqrt[4]{4c}}{|d|}\)

Try it.

ⓐ \(\sqrt{\frac{72{x}^{5}}{{y}^{6}}}\) ⓑ \(\sqrt[5]{\frac{96{r}^{11}}{{s}^{5}}}\) ⓒ \(\sqrt[6]{\frac{128{u}^{7}}{{v}^{12}}}\)

Try it.

ⓐ \(\sqrt{\frac{28{p}^{7}}{{q}^{2}}}\) ⓑ \(\sqrt[3]{\frac{81{s}^{8}}{{t}^{3}}}\) ⓒ \(\sqrt[4]{\frac{64{p}^{15}}{{q}^{12}}}\)

Solution

ⓐ \(\frac{2|{p}^{3}|\sqrt{7p}}{|q|}\) ⓑ \(\frac{3{s}^{2}\sqrt[3]{3{s}^{2}}}{t}\)
ⓒ \(\frac{2|{p}^{3}|\sqrt[4]{4{p}^{3}}}{|{q}^{3}|}\)

Try it.

ⓐ \(\sqrt{\frac{45{r}^{3}}{{s}^{10}}}\) ⓑ \(\sqrt[3]{\frac{625{u}^{10}}{{v}^{3}}}\) ⓒ \(\sqrt[4]{\frac{729{c}^{21}}{{d}^{8}}}\)

Try it.

ⓐ \(\sqrt{\frac{32{x}^{5}{y}^{3}}{18{x}^{3}y}}\) ⓑ \(\sqrt[3]{\frac{5{x}^{6}{y}^{9}}{40{x}^{5}{y}^{3}}}\) ⓒ \(\sqrt[4]{\frac{5{a}^{8}{b}^{6}}{80{a}^{3}{b}^{2}}}\)

Solution

ⓐ \(\frac{4|xy|}{3}\) ⓑ \(\frac{{y}^{2}\sqrt[3]{x}}{2}\) ⓒ \(\frac{|ab|\sqrt[4]{a}}{2}\)

Try it.

ⓐ \(\sqrt{\frac{75{r}^{6}{s}^{8}}{48r{s}^{4}}}\) ⓑ \(\sqrt[3]{\frac{24{x}^{8}{y}^{4}}{81{x}^{2}y}}\) ⓒ \(\sqrt[4]{\frac{32{m}^{9}{n}^{2}}{162m{n}^{2}}}\)

Try it.

ⓐ \(\sqrt{\frac{27{p}^{2}q}{108{p}^{4}{q}^{3}}}\) ⓑ \(\sqrt[3]{\frac{16{c}^{5}{d}^{7}}{250{c}^{2}{d}^{2}}}\) ⓒ \(\sqrt[6]{\frac{2{m}^{9}{n}^{7}}{128{m}^{3}n}}\)

Solution

ⓐ \(\frac{1}{2|pq|}\) ⓑ \(\frac{2cd\sqrt[3]{{d}^{2}}}{5}\)
ⓒ \(\frac{|mn|}{2}\)

Try it.

ⓐ \(\sqrt{\frac{50{r}^{5}{s}^{2}}{128{r}^{2}{s}^{6}}}\) ⓑ \(\sqrt[3]{\frac{24{m}^{9}{n}^{7}}{375{m}^{4}{n}^{}}}\) ⓒ \(\sqrt[4]{\frac{81{m}^{2}{n}^{8}}{256{m}^{1}{n}^{2}}}\)

Try it.

ⓐ \(\frac{\sqrt{45{p}^{9}}}{\sqrt{5{q}^{2}}}\) ⓑ \(\frac{\sqrt[4]{64}}{\sqrt[4]{2}}\) ⓒ \(\frac{\sqrt[5]{128{x}^{8}}}{\sqrt[5]{2{x}^{2}}}\)

Solution

ⓐ \(\frac{3{p}^{4}\sqrt{p}}{|q|}\) ⓑ \(2\sqrt[4]{2}\)
ⓒ \(2x\sqrt[5]{2x}\)

Try it.

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

Try it.

ⓐ \(\frac{\sqrt{50{m}^{7}}}{\sqrt{2m}}\) ⓑ \(\sqrt[3]{\frac{1250}{2}}\) ⓒ \(\sqrt[4]{\frac{486{y}^{9}}{2{y}^{3}}}\)

Solution

ⓐ \(5|{m}^{3}|\) ⓑ \(5\sqrt[3]{5}\)
ⓒ \(3|y|\sqrt[4]{3{y}^{2}}\)

Try it.

ⓐ \(\frac{\sqrt{72{n}^{11}}}{\sqrt{2n}}\) ⓑ \(\sqrt[3]{\frac{162}{6}}\) ⓒ \(\sqrt[4]{\frac{160{r}^{10}}{5{r}^{3}}}\)

Condensed — the full section is in OpenStax Intermediate 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: \(\frac{{x}^{9}}{{x}^{4}}.\)
If you missed this problem, review .

Жауап беріңіз

\({x}^{5}\)
Simplify: \(\frac{{y}^{3}}{{y}^{11}}.\)
If you missed this problem, review .

Жауап беріңіз

\(\frac{1}{{y}^{8}}\)
Simplify: \({({n}^{2})}^{6}.\)
If you missed this problem, review .

Жауап беріңіз

\({n}^{12}\)
Simplify: \(\sqrt{98}.\)
Simplify: \(\sqrt{48}.\)

Жауап беріңіз

\(4\sqrt{3}\)
Simplify: \(\sqrt{45}.\)

Жауап беріңіз

\(3\sqrt{5}\)

Simplify: ⓐ \(\sqrt{500}\) ⓑ \(\sqrt[3]{16}\) ⓒ \(\sqrt[4]{243}.\)

Жауап беріңіз

ⓐ

	\(\sqrt{500}\)
Rewrite the radicand as a product using the largest perfect square factor.	\(\sqrt{100\cdot 5}\)
Rewrite the radical as the product of two radicals.	\(\sqrt{100}\cdot \sqrt{5}\)
Simplify.	\(10\sqrt{5}\)

ⓑ

	\(\sqrt[3]{16}\)
Rewrite the radicand as a product using the greatest perfect cube factor. \({2}^{3}=8\)	\(\sqrt[3]{8\cdot 2}\)
Rewrite the radical as the product of two radicals.	\(\sqrt[3]{8}\cdot \sqrt[3]{{2}^{}}\)
Simplify.	\(2\ \sqrt[3]{{2}^{}}\)

ⓒ

	\(\sqrt[4]{243}\)
Rewrite the radicand as a product using the greatest perfect fourth power factor. \({3}^{4}=81\)	\(\sqrt[4]{81\cdot {3}^{}}\)
Rewrite the radical as the product of two radicals.	\(\sqrt[4]{81}\cdot \sqrt[4]{{3}^{}}\)
Simplify.	\(3\ \sqrt[4]{{3}^{}}\)

Simplify: ⓐ \(\sqrt{288}\) ⓑ \(\sqrt[3]{81}\) ⓒ \(\sqrt[4]{64}.\)

Жауап беріңіз

ⓐ \(12\sqrt{2}\) ⓑ \(3\sqrt[3]{3}\) ⓒ \(2\sqrt[4]{4}\)
Simplify: ⓐ \(\sqrt{432}\) ⓑ \(\sqrt[3]{625}\) ⓒ \(\sqrt[4]{729}.\)

Жауап беріңіз

ⓐ \(12\sqrt{3}\) ⓑ \(5\sqrt[3]{5}\) ⓒ \(3\sqrt[4]{9}\)

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

Жауап беріңіз

ⓐ

	\(\sqrt{{x}^{3}}\)
Rewrite the radicand as a product using the largest perfect square factor.	\(\sqrt{{x}^{2}\cdot x}\)
Rewrite the radical as the product of two radicals.	\(\sqrt{{x}^{2}}\cdot \sqrt{x}\)
Simplify.	\(\|x\|\ \sqrt{x}\)

ⓑ

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

Жауап беріңіз

ⓐ \({b}^{2}\sqrt{b}\) ⓑ \(|y|\sqrt[4]{{y}^{2}}\) ⓒ \(z\sqrt[3]{{z}^{2}}\)
Simplify: ⓐ \(\sqrt{{p}^{9}}\) ⓑ \(\sqrt[5]{{y}^{8}}\) ⓒ \(\sqrt[6]{{q}^{13}}\)

Жауап беріңіз

ⓐ \({p}^{4}\sqrt{p}\) ⓑ \(y\sqrt[5]{{y}^{3}}\)
ⓒ \({q}^{2}\sqrt[6]{q}\)

Жауап беріңіз

ⓐ

	\(\sqrt{72{n}^{7}}\)
Rewrite the radicand as a product using the largest perfect square factor.	\(\sqrt{36{n}^{6}\cdot 2n}\)
Rewrite the radical as the product of two radicals.	\(\sqrt{36{n}^{6}}\cdot \sqrt{2n}\)
Simplify.	\(6\|{n}^{3}\|\ \sqrt{2n}\)

ⓑ

	\(\sqrt[3]{24{x}^{7}}\)
Rewrite the radicand as a product using perfect cube factors.	\(\sqrt[3]{8{x}^{6}\cdot 3x}\)
Rewrite the radical as the product of two radicals.	\(\sqrt[3]{8{x}^{6}}\cdot \sqrt[3]{{3}^{}x}\)
Rewrite the first radicand as \({(2{x}^{2})}^{3}.\)	\(\sqrt[3]{{({2}^{}{x}^{2})}^{3}}\cdot \sqrt[3]{{3}^{}x}\)
Simplify.	\(2{x}^{2}\ \sqrt[3]{{3}^{}x}\)

ⓒ

	\(\sqrt[4]{80{y}^{14}}\)
Rewrite the radicand as a product using perfect fourth power factors.	\(\sqrt[4]{16{y}^{12}\cdot {5}^{}{y}^{2}}\)
Rewrite the radical as the product of two radicals.	\(\sqrt[4]{16{y}^{12}}\cdot \sqrt[4]{5{y}^{2}{}^{}}\)
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: ⓐ \(\sqrt{32{y}^{5}}\) ⓑ \(\sqrt[3]{54{p}^{10}}\) ⓒ \(\sqrt[4]{64{q}^{10}}.\)

Жауап беріңіз

ⓐ \(4{y}^{2}\sqrt{2y}\) ⓑ \(3{p}^{3}\sqrt[3]{2p}\)
ⓒ \(2{q}^{2}\sqrt[4]{4{q}^{2}}\)
Simplify: ⓐ \(\sqrt{75{a}^{9}}\) ⓑ \(\sqrt[3]{128{m}^{11}}\) ⓒ \(\sqrt[4]{162{n}^{7}}.\)

Жауап беріңіз

ⓐ \(5{a}^{4}\sqrt{3a}\) ⓑ \(4{m}^{3}\sqrt[3]{2{m}^{2}}\)
ⓒ \(3|n|\sqrt[4]{2{n}^{3}}\)

Жауап беріңіз

ⓐ

	\(\sqrt{63{u}^{3}{v}^{5}}\)
Rewrite the radicand as a product using the largest perfect square factor.	\(\sqrt{9{u}^{2}{v}^{4}\cdot 7uv}\)
Rewrite the radical as the product of two radicals.	\(\sqrt{9{u}^{2}{v}^{4}}\cdot \sqrt{7uv}\)
Rewrite the first radicand as \({(3u{v}^{2})}^{2}.\)	\(\sqrt{{(3u{v}^{2})}^{2}}\cdot \sqrt{7uv}\)
Simplify.	\(3\|u\|{v}^{2}\ \sqrt{7uv}\)

ⓑ

	\(\sqrt[3]{40{x}^{4}{y}^{5}}\)
Rewrite the radicand as a product using the largest perfect cube factor.	\(\sqrt[3]{8{x}^{3}{y}^{3}\cdot 5x{y}^{2}}\)
Rewrite the radical as the product of two radicals.	\(\sqrt[3]{8{x}^{3}{y}^{3}}\cdot \sqrt[3]{5x{y}^{2}}\)
Rewrite the first radicand as \({(2xy)}^{3}.\)	\(\sqrt[3]{{(2xy)}^{3}}\cdot \sqrt[3]{5x{y}^{2}}\)
Simplify.	\(2xy\ \sqrt[3]{5x{y}^{2}}\)

ⓒ

	\(\sqrt[4]{48{x}^{4}{y}^{7}}\)
Rewrite the radicand as a product using the largest perfect fourth power factor.	\(\sqrt[4]{16{x}^{4}{y}^{4}\cdot 3{y}^{3}}\)
Rewrite the radical as the product of two radicals.	\(\sqrt[4]{16{x}^{4}{y}^{4}}\cdot \sqrt[4]{3{y}^{3}}\)
Rewrite the first radicand as \({(2xy)}^{4}.\)	\(\sqrt[4]{{(2xy)}^{4}}\cdot \sqrt[4]{3{y}^{3}}\)
Simplify.	\(2\|xy\|\ \sqrt[4]{3{y}^{3}}\)

Simplify: ⓐ \(\sqrt{98{a}^{7}{b}^{5}}\) ⓑ \(\sqrt[3]{56{x}^{5}{y}^{4}}\) ⓒ \(\sqrt[4]{32{x}^{5}{y}^{8}}.\)

Жауап беріңіз

ⓐ \(7|{a}^{3}|{b}^{2}{\sqrt{2ab}}^{}\)
ⓑ \(2xy\sqrt[3]{7{x}^{2}y}\) ⓒ \(2|x|{y}^{2}\sqrt[4]{2x}\)
Simplify: ⓐ \(\sqrt{180{m}^{9}{n}^{11}}\) ⓑ \(\sqrt[3]{72{x}^{6}{y}^{5}}\) ⓒ \(\sqrt[4]{80{x}^{7}{y}^{4}}.\)

Жауап беріңіз

ⓐ \(6{m}^{4}|{n}^{5}|\sqrt{5mn}\)
ⓑ \(2{x}^{2}y\sqrt[3]{9{y}^{2}}\) ⓒ \(2|xy|\sqrt[4]{5{x}^{3}}\)
Simplify: ⓐ \(\sqrt[3]{-27}\) ⓑ \(\sqrt[4]{-16}.\)

Жауап беріңіз

ⓐ

\(\sqrt[3]{-27}\)
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: ⓐ \(\sqrt[3]{-64}\) ⓑ \(\sqrt[4]{-81}.\)

Жауап беріңіз

ⓐ \(-4\) ⓑ \(\text{no real number}\)
Simplify: ⓐ \(\sqrt[3]{-625}\) ⓑ \(\sqrt[4]{-324}.\)

Жауап беріңіз

ⓐ \(-5\sqrt[3]{5}\) ⓑ no real number

Simplify: ⓐ \(3+\sqrt{32}\) ⓑ \(\frac{4-\sqrt{48}}{2}.\)

Жауап беріңіз

ⓐ

	\(3+\sqrt{32}\)
Rewrite the radicand as a product using the largest perfect square factor.	\(3+\sqrt{16\cdot 2}\)
Rewrite the radical as the product of two radicals.	\(3+\sqrt{16}\cdot \sqrt{2}\)
Simplify.	\(3+4\sqrt{2}\)

The terms cannot be added as one has a radical and the other does not. Trying to add an integer and a radical is like trying to add an integer and a variable. They are not like terms!

ⓑ

	\(\frac{4-\sqrt{48}}{2}\)
Rewrite the radicand as a product using the largest perfect square factor.	\(\frac{4-\sqrt{16\cdot 3}}{2}\)
Rewrite the radical as the product of two radicals.	\(\frac{4-\sqrt{16}\cdot \sqrt{3}}{2}\)
Simplify.	\(\frac{4-4\sqrt{3}}{2}\)
Factor the common factor from the numerator.	\(\frac{4(1-\sqrt{3})}{2}\)
Remove the common factor, 2, from the numerator and denominator.	\(\frac{2\cdot 2(1-\sqrt{3})}{2}\)
Simplify.	\(2(1-\sqrt{3})\)

Simplify: ⓐ \(5+\sqrt{75}\) ⓑ \(\frac{10-\sqrt{75}}{5}\)

Жауап беріңіз

ⓐ \(5+5\sqrt{3}\) ⓑ \(2-\sqrt{3}\)
Simplify: ⓐ \(2+\sqrt{98}\) ⓑ \(\frac{6-\sqrt{45}}{3}\)

Жауап беріңіз

ⓐ \(2+7\sqrt{2}\) ⓑ \(2-\sqrt{5}\)

Жауап беріңіз

ⓐ

	\(\sqrt{\frac{45}{80}}\)
Simplify inside the radical first.
Rewrite showing the common factors of the numerator and denominator.	\(\sqrt{\frac{5\cdot 9}{5\cdot 16}}\)
Simplify the fraction by removing common factors.	\(\sqrt{\frac{9}{16}}\)
Simplify. Note \({(\frac{3}{4})}^{2}=\frac{9}{16}.\)	\(\frac{3}{4}\)

ⓑ

	\(\sqrt[3]{\frac{16}{54}}\)
Simplify inside the radical first.
Rewrite showing the common factors of the numerator and denominator.	\(\sqrt[3]{\frac{2\cdot 8}{2\cdot 27}}\)
Simplify the fraction by removing common factors.	\(\sqrt[3]{\frac{8}{27}}\)
Simplify. Note \({(\frac{2}{3})}^{3}=\frac{8}{27}.\)	\(\frac{2}{3}\)

ⓒ

	\(\sqrt[4]{\frac{5}{80}}\)
Simplify inside the radical first.
Rewrite showing the common factors of the numerator and denominator.	\(\sqrt[4]{\frac{5\cdot 1}{5\cdot 16}}\)
Simplify the fraction by removing common factors.	\(\sqrt[4]{\frac{1}{16}}\)
Simplify. Note \({(\frac{1}{2})}^{4}=\frac{1}{16}.\)	\(\frac{1}{2}\)

Simplify: ⓐ \(\sqrt{\frac{75}{48}}\) ⓑ \(\sqrt[3]{\frac{54}{250}}\) ⓒ \(\sqrt[4]{\frac{32}{162}}.\)

Жауап беріңіз

ⓐ \(\frac{5}{4}\) ⓑ \(\frac{3}{5}\) ⓒ \(\frac{2}{3}\)
Simplify: ⓐ \(\sqrt{\frac{98}{162}}\) ⓑ \(\sqrt[3]{\frac{24}{375}}\) ⓒ \(\sqrt[4]{\frac{4}{324}}.\)

Жауап беріңіз

ⓐ \(\frac{7}{9}\) ⓑ \(\frac{2}{5}\) ⓒ \(\frac{1}{3}\)

Simplify: ⓐ \(\sqrt{\frac{{m}^{6}}{{m}^{4}}}\) ⓑ \(\sqrt[3]{\frac{{a}^{8}}{{a}^{5}}}\) ⓒ \(\sqrt[4]{\frac{{a}^{10}}{{a}^{2}}}.\)

Жауап беріңіз

ⓐ

	\(\sqrt{\frac{{m}^{6}}{{m}^{4}}}\)
Simplify the fraction inside the radical first.
Divide the like bases by subtracting the exponents.	\(\sqrt{{m}^{2}}\)
Simplify.	\(\|m\|\)

ⓑ

	\(\sqrt[3]{\frac{{a}^{8}}{{a}^{5}}}\)
Use the Quotient Property of exponents to simplify the fraction under the radical first.	\(\sqrt[3]{{a}^{3}}\)
Simplify.	\(a\)

ⓒ

	\(\sqrt[4]{\frac{{a}^{10}}{{a}^{2}}}\)
Use the Quotient Property of exponents to 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: ⓐ \(\sqrt{\frac{{a}^{8}}{{a}^{6}}}\) ⓑ \(\sqrt[4]{\frac{{x}^{7}}{{x}^{3}}}\) ⓒ \(\sqrt[4]{\frac{{y}^{17}}{{y}^{5}}}.\)

Жауап беріңіз

ⓐ \(|a|\) ⓑ \(|x|\) ⓒ \({y}^{3}\)
Simplify: ⓐ \(\sqrt{\frac{{x}^{14}}{{x}^{10}}}\) ⓑ \(\sqrt[3]{\frac{{m}^{13}}{{m}^{7}}}\) ⓒ \(\sqrt[5]{\frac{{n}^{12}}{{n}^{2}}}.\)

Жауап беріңіз

ⓐ \({x}^{2}\) ⓑ \({m}^{2}\) ⓒ \({n}^{2}\)
Simplify: \(\sqrt{\frac{27{m}^{3}}{196}}.\)
Simplify: \(\sqrt{\frac{24{p}^{3}}{49}}.\)

Жауап беріңіз

\(\frac{2|p|\sqrt{6p}}{7}\)
Simplify: \(\sqrt{\frac{48{x}^{5}}{100}}.\)

Жауап беріңіз

\(\frac{2{x}^{2}\sqrt{3x}}{5}\)

Simplify: ⓐ \(\sqrt{\frac{45{x}^{5}}{{y}^{4}}}\) ⓑ \(\sqrt[3]{\frac{24{x}^{7}}{{y}^{3}}}\) ⓒ \(\sqrt[4]{\frac{48{x}^{10}}{{y}^{8}}}.\)

Жауап беріңіз

ⓐ

	\(\sqrt{\frac{45{x}^{5}}{{y}^{4}}}\)
We cannot simplify the fraction in the radicand. Rewrite using the Quotient Property.	\(\frac{\sqrt{45{x}^{5}}}{\sqrt{{y}^{4}}}\)
Simplify the radicals in the numerator and the denominator.	\(\frac{\sqrt{9{x}^{4}}\cdot \sqrt{5x}}{{y}^{2}}\)
Simplify.	\(\frac{3{x}^{2}\sqrt{5x}}{{y}^{2}}\)

ⓑ

	\(\sqrt[3]{\frac{24{x}^{7}}{{y}^{3}}}\)
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.	\(\frac{\sqrt[3]{{(2{x}^{2})}^{3}}\cdot \sqrt[3]{3x}}{\sqrt[3]{{y}^{3}}}\)
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.	\(\frac{\sqrt[4]{48{x}^{10}}}{\sqrt[4]{{y}^{8}}}\)
Use the Quotient Property to write as two radicals. 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.	\(\frac{\sqrt[4]{{(2{x}^{2})}^{4}}\cdot \sqrt[4]{3{x}^{2}}}{\sqrt[4]{{({y}^{2})}^{4}}}\)
Simplify.	\(\frac{2{x}^{2}\sqrt[4]{3{x}^{2}}}{{y}^{2}}\)

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

Жауап беріңіз

ⓐ \(\frac{4|m|\sqrt{5m}}{|{n}^{3}|}\) ⓑ \(\frac{3{c}^{3}\sqrt[3]{4c}}{{d}^{2}}\)
ⓒ \(\frac{2{x}^{2}\sqrt[4]{5{x}^{2}}}{|y|}\)
Simplify: ⓐ \(\sqrt{\frac{54{u}^{7}}{{v}^{8}}}\) ⓑ \(\sqrt[3]{\frac{40{r}^{3}}{{s}^{6}}}\) ⓒ \(\sqrt[4]{\frac{162{m}^{14}}{{n}^{12}}}.\)

Жауап беріңіз

ⓐ \(\frac{3{u}^{3}\sqrt{6u}}{{v}^{4}}\) ⓑ \(\frac{2r\sqrt[3]{5}}{{s}^{2}}\)
ⓒ \(\frac{3|{m}^{3}|\sqrt[4]{2{m}^{2}}}{|{n}^{3}|}\)

Simplify: ⓐ \(\sqrt{\frac{18{p}^{5}{q}^{7}}{32p{q}^{2}}}\) ⓑ \(\sqrt[3]{\frac{16{x}^{5}{y}^{7}}{54{x}^{2}{y}^{2}}}\) ⓒ \(\sqrt[4]{\frac{5{a}^{8}{b}^{6}}{80{a}^{3}{b}^{2}}}.\)

Жауап беріңіз

ⓐ

	\(\sqrt{\frac{18{p}^{5}{q}^{7}}{32p{q}^{2}}}\)
Simplify the fraction in the radicand, if possible.	\(\sqrt{\frac{9{p}^{4}{q}^{5}}{16}}\)
Rewrite using the Quotient Property.	\(\frac{\sqrt{9{p}^{4}{q}^{5}}}{\sqrt{16}}\)
Simplify the radicals in the numerator and the denominator.	\(\frac{\sqrt{9{p}^{4}{q}^{4}}\cdot \sqrt{q}}{4}\)
Simplify.	\(\frac{3{p}^{2}{q}^{2}\sqrt{q}}{4}\)

ⓑ

	\(\sqrt[3]{\frac{16{x}^{5}{y}^{7}}{54{x}^{2}{y}^{2}}}\)
Simplify the fraction in the radicand, if possible.	\(\sqrt[3]{\frac{8{x}^{3}{y}^{5}}{27}}\)
Rewrite using the Quotient Property.	\(\frac{\sqrt[3]{8{x}^{3}{y}^{5}}}{\sqrt[3]{27}}\)
Simplify the radicals in the numerator and the denominator.	\(\frac{\sqrt[3]{8{x}^{3}{y}^{3}}\cdot \sqrt[3]{{y}^{2}}}{\sqrt[3]{27}}\)
Simplify.	\(\frac{2xy\ \sqrt[3]{{y}^{2}}}{3}\)

ⓒ

	\(\sqrt[4]{\frac{5{a}^{8}{b}^{6}}{80{a}^{3}{b}^{2}}}\)
Simplify the fraction in the radicand, if possible.	\(\sqrt[4]{\frac{{a}^{5}{b}^{4}}{16}}\)
Rewrite using the Quotient Property.	\(\frac{\sqrt[4]{{a}^{5}{b}^{4}}}{\sqrt[4]{16}}\)
Simplify the radicals in the numerator and the denominator.	\(\frac{\sqrt[4]{{a}^{4}{b}^{4}}\cdot \sqrt[4]{a}}{\sqrt[4]{16}}\)
Simplify.	\(\frac{\|ab\|\ \sqrt[4]{a}}{2}\)

Simplify: ⓐ \(\sqrt{\frac{50{x}^{5}{y}^{3}}{72{x}^{4}y}}\) ⓑ \(\sqrt[3]{\frac{16{x}^{5}{y}^{7}}{54{x}^{2}{y}^{2}}}\) ⓒ \(\sqrt[4]{\frac{5{a}^{8}{b}^{6}}{80{a}^{3}{b}^{2}}}.\)

Жауап беріңіз

ⓐ \(\frac{5|y|\sqrt{x}}{6}\) ⓑ \(\frac{2xy\sqrt[3]{{y}^{2}}}{3}\)
ⓒ \(\frac{|ab|\sqrt[4]{a}}{2}\)
Simplify: ⓐ \(\sqrt{\frac{48{m}^{7}{n}^{2}}{100{m}^{5}{n}^{8}}}\) ⓑ \(\sqrt[3]{\frac{54{x}^{7}{y}^{5}}{250{x}^{2}{y}^{2}}}\) ⓒ \(\sqrt[4]{\frac{32{a}^{9}{b}^{7}}{162{a}^{3}{b}^{3}}}.\)

Жауап беріңіз

ⓐ \(\frac{2|m|\sqrt{3}}{5|{n}^{3}|}\) ⓑ \(\frac{3xy\sqrt[3]{{x}^{2}}}{5}\)
ⓒ \(\frac{2|ab|\sqrt[4]{{a}^{2}}}{3}\)

Simplify: ⓐ \(\frac{\sqrt{48{a}^{7}}}{\sqrt{3a}}\) ⓑ \(\frac{\sqrt[3]{-108}}{\sqrt[3]{2}}\) ⓒ \(\frac{\sqrt[4]{96{x}^{7}}}{\sqrt[4]{3{x}^{2}}}.\)

Жауап беріңіз

ⓐ

	\(\frac{\sqrt{48{a}^{7}}}{\sqrt{3a}}\)
The denominator cannot be simplified, so use the Quotient Property to write as one radical.	\(\sqrt{\frac{48{a}^{7}}{3a}}\)
Simplify the fraction under the radical.	\(\sqrt{16{a}^{6}}\)
Simplify.	\(4\|{a}^{3}\|\)

ⓑ

	\(\frac{\sqrt[3]{-108}}{\sqrt[3]{2}}\)
The denominator cannot be simplified, 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.	\(-3\ \sqrt[3]{2}\)

ⓒ

	\(\frac{\sqrt[4]{96{x}^{7}}}{\sqrt[4]{3{x}^{2}}}\)
The denominator cannot be simplified, 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.	\(\sqrt[4]{16{x}^{4}}\cdot \sqrt[4]{2x}\)
Rewrite the radical as the product of two radicals.	\(\sqrt[4]{{(2x)}^{4}}\cdot \sqrt[4]{2x}\)
Simplify.	\(2\|x\|\ \sqrt[4]{2x}\)

Symbols used here

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

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

\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.

|x|

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

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: Simplify Radical Expressions

Use the Product Property to simplify radical expressions
Use the Quotient Property to simplify radical expressions
Find the largest factor in the radicand that is a perfect power of the index. Rewrite the radicand as a product of two factors, using that factor.
Use the product rule to rewrite the radical as the product of two radicals.
Simplify the root of the perfect power.
Simplify the fraction in the radicand, if possible.
Use the Quotient Property to rewrite the radical as the quotient of two radicals.
Simplify the radicals in the numerator and the denominator.

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.

Өзіңіздіңіңізді сынап көріңіз

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

Келесіде Algebra

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	\(\sqrt[3]{-27}\)
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.