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.

Now you Không máy tính nào có thể tính toán được nó, nhưng các mảnh của nó có thể tính toán được. Thử một trong những cách dưới đây, hoặc gõ theo ý bạn.

⌨ Kiểu ✎ Vẽ đi.

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π Giải quyết

Luyện tập (40)

Đầu tiên thử mỗi câu trên giấy. Cho thấy câu trả lời để kiểm tra; những câu trả lời đã kiểm tra có thể mở trong trình giải cho mỗi bước.

1. Simplify: \(\frac{{x}^{9}}{{x}^{4}}.\)

   Làm việc đi. Gợi ý

   Giải đáp

   \({x}^{5}\)

2. Simplify: \(\frac{{y}^{3}}{{y}^{11}}.\)

   Làm việc đi. Gợi ý

   Giải đáp

   \(\frac{1}{{y}^{8}}\)

3. Simplify: \({({n}^{2})}^{6}.\)

   Làm việc đi. Gợi ý

   Giải đáp

   \({n}^{12}\)

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

   Làm việc đi. Gợi ý

5. Simplify: \(\sqrt{48}.\)

   Làm việc đi. Gợi ý

   Giải đáp

   \(4\sqrt{3}\)

6. Simplify: \(\sqrt{45}.\)

   Làm việc đi. Gợi ý

   Giải đáp

   \(3\sqrt{5}\)

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

   Làm việc đi. Gợi ý

   Giải đáp

   ⓐ
   

   	\(\sqrt{500}\)
   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.	\(3\ \sqrt[4]{{3}^{}}\)

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

   Làm việc đi. Gợi ý

   Giải đáp

   ⓐ \(12\sqrt{2}\) ⓑ \(3\sqrt[3]{3}\) ⓒ \(2\sqrt[4]{4}\)

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

   Làm việc đi. Gợi ý

   Giải đáp

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

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

    Làm việc đi. Gợi ý

    Giải đáp

    ⓐ
    

    	\(\sqrt{{x}^{3}}\)
    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.	\(|x|\ \sqrt[4]{{x}^{3}}\)

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

    Làm việc đi. Gợi ý

    Giải đáp

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

12. Simplify: ⓐ \(\sqrt{{p}^{9}}\) ⓑ \(\sqrt[5]{{y}^{8}}\) ⓒ \(\sqrt[6]{{q}^{13}}\)

    Làm việc đi. Gợi ý

    Giải đáp

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

13. Simplify: ⓐ \(\sqrt{72{n}^{7}}\) ⓑ \(\sqrt[3]{24{x}^{7}}\) ⓒ \(\sqrt[4]{80{y}^{14}}.\)

    Làm việc đi. Gợi ý

    Giải đáp

    ⓐ
    

    	\(\sqrt{72{n}^{7}}\)
    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.	\(2|{y}^{3}|\ \sqrt[4]{{5}^{}{y}^{2}}\)

14. Simplify: ⓐ \(\sqrt{32{y}^{5}}\) ⓑ \(\sqrt[3]{54{p}^{10}}\) ⓒ \(\sqrt[4]{64{q}^{10}}.\)

    Làm việc đi. Gợi ý

    Giải đáp

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

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

    Làm việc đi. Gợi ý

    Giải đáp

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

16. Simplify: ⓐ \(\sqrt{63{u}^{3}{v}^{5}}\) ⓑ \(\sqrt[3]{40{x}^{4}{y}^{5}}\) ⓒ \(\sqrt[4]{48{x}^{4}{y}^{7}}.\)

    Làm việc đi. Gợi ý

    Giải đáp

    ⓐ
    

    	\(\sqrt{63{u}^{3}{v}^{5}}\)
    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.	\(2|xy|\ \sqrt[4]{3{y}^{3}}\)

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

    Làm việc đi. Gợi ý

    Giải đáp

    ⓐ \(7|{a}^{3}|{b}^{2}{\sqrt{2ab}}^{}\)
    ⓑ \(2xy\sqrt[3]{7{x}^{2}y}\) ⓒ \(2|x|{y}^{2}\sqrt[4]{2x}\)

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

    Làm việc đi. Gợi ý

    Giải đáp

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

19. Simplify: ⓐ \(\sqrt[3]{-27}\) ⓑ \(\sqrt[4]{-16}.\)

    Làm việc đi. Gợi ý

    Giải đáp

    ⓐ
    

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

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

    Làm việc đi. Gợi ý

    Giải đáp

    ⓐ \(-4\) ⓑ \(\text{no real number}\)

21. Simplify: ⓐ \(\sqrt[3]{-625}\) ⓑ \(\sqrt[4]{-324}.\)

    Làm việc đi. Gợi ý

    Giải đáp

    ⓐ \(-5\sqrt[3]{5}\) ⓑ no real number

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

    Làm việc đi. Gợi ý

    Giải đáp

    ⓐ
    

    	\(3+\sqrt{32}\)
    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.	\(2(1-\sqrt{3})\)

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

    Làm việc đi. Gợi ý

    Giải đáp

    ⓐ \(5+5\sqrt{3}\) ⓑ \(2-\sqrt{3}\)

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

    Làm việc đi. Gợi ý

    Giải đáp

    ⓐ \(2+7\sqrt{2}\) ⓑ \(2-\sqrt{5}\)

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

    Làm việc đi. Gợi ý

    Giải đáp

    ⓐ
    

    	\(\sqrt{\frac{45}{80}}\)
    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 \({(\frac{1}{2})}^{4}=\frac{1}{16}.\)	\(\frac{1}{2}\)

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

    Làm việc đi. Gợi ý

    Giải đáp

    ⓐ \(\frac{5}{4}\) ⓑ \(\frac{3}{5}\) ⓒ \(\frac{2}{3}\)

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

    Làm việc đi. Gợi ý

    Giải đáp

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

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

    Làm việc đi. Gợi ý

    Giải đáp

    ⓐ
    

    	\(\sqrt{\frac{{m}^{6}}{{m}^{4}}}\)
    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.	\({a}^{2}\)

29. Simplify: ⓐ \(\sqrt{\frac{{a}^{8}}{{a}^{6}}}\) ⓑ \(\sqrt[4]{\frac{{x}^{7}}{{x}^{3}}}\) ⓒ \(\sqrt[4]{\frac{{y}^{17}}{{y}^{5}}}.\)

    Làm việc đi. Gợi ý

    Giải đáp

    ⓐ \(|a|\) ⓑ \(|x|\) ⓒ \({y}^{3}\)

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

    Làm việc đi. Gợi ý

    Giải đáp

    ⓐ \({x}^{2}\) ⓑ \({m}^{2}\) ⓒ \({n}^{2}\)

31. Simplify: \(\sqrt{\frac{27{m}^{3}}{196}}.\)

    Làm việc đi. Gợi ý

32. Simplify: \(\sqrt{\frac{24{p}^{3}}{49}}.\)

    Làm việc đi. Gợi ý

    Giải đáp

    \(\frac{2|p|\sqrt{6p}}{7}\)

33. Simplify: \(\sqrt{\frac{48{x}^{5}}{100}}.\)

    Làm việc đi. Gợi ý

    Giải đáp

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

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

    Làm việc đi. Gợi ý

    Giải đáp

    ⓐ
    

    	\(\sqrt{\frac{45{x}^{5}}{{y}^{4}}}\)
    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.	\(\frac{2{x}^{2}\sqrt[4]{3{x}^{2}}}{{y}^{2}}\)

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

    Làm việc đi. Gợi ý

    Giải đáp

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

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

    Làm việc đi. Gợi ý

    Giải đáp

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

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

    Làm việc đi. Gợi ý

    Giải đáp

    ⓐ
    

    	\(\sqrt{\frac{18{p}^{5}{q}^{7}}{32p{q}^{2}}}\)
    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.	\(\frac{|ab|\ \sqrt[4]{a}}{2}\)

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

    Làm việc đi. Gợi ý

    Giải đáp

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

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

    Làm việc đi. Gợi ý

    Giải đáp

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

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

    Làm việc đi. Gợi ý

    Giải đáp

    ⓐ
    

    	\(\frac{\sqrt{48{a}^{7}}}{\sqrt{3a}}\)
    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.	\(2|x|\ \sqrt[4]{2x}\)

Để cho anh làm việc

Một tài khoản miễn phí thêm ghi chú vào mỗi bài học, ghi chép những gì bạn đã hoàn thành, các bài toán bạn giải quyết ở một nơi, và một giáo viên bạn có thể hỏi về trang này. Toán học là mở cho mọi người, dù có đăng nhập hay không.

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Nhấn vào bất kỳ ký hiệu nào để xem định nghĩa đầy đủ, một bức ảnh, và ý nghĩa của từng chữ cái trong nó.

\sqrt{x}square root, n-th root
The non-negative number whose square (n-th power) is x.\leqless/greater than or equal
Inequalities that allow equality; < and > exclude it.\neqnot equal
The two sides are different.\pmplus 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.iimaginary unit
i² = −1.\log_b xlogarithm, natural log
The exponent b must be raised to for x; ln uses base e.\mathbb{N}number sets
Naturals, integers, rationals, reals, complex numbers.

Làm thế nào: Simplify Radical Expressions

1. Use the Product Property to simplify radical expressions
2. Use the Quotient Property to simplify radical expressions
3. 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.
4. Use the product rule to rewrite the radical as the product of two radicals.
5. Simplify the root of the perfect power.
6. Simplify the fraction in the radicand, if possible.
7. Use the Quotient Property to rewrite the radical as the quotient of two radicals.
8. Simplify the radicals in the numerator and the denominator.

Câu hỏi người ta hỏi

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.

Một số phần của trang này được chuyển từ OpenStax Intermediate Algebra 2e (CC BY-NC-SA 4.0). Được nén lại và giải thích lại ở đây; lỗi là của chúng tôi. Bởi vì nguồn của nó được cấp phép cho sử dụng không thương mại, bài học này là miễn phí, không bán gì, và được chia sẻ dưới cùng một giấy phép.

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