Algebra Examples

$\sqrt{\frac{\sqrt[5]{x^{4}} \sqrt[4]{y^{3}}}{\sqrt[3]{x^{2}}}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite the expression using the least common index of $20$ .

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Step 1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{4}}$ as $x^{\frac{4}{5}}$ .

$\sqrt{\frac{x^{\frac{4}{5}} \sqrt[4]{y^{3}}}{\sqrt[3]{x^{2}}}}$

Step 1.1.2

Rewrite $x^{\frac{4}{5}}$ as $x^{\frac{16}{20}}$ .

$\sqrt{\frac{x^{\frac{16}{20}} \sqrt[4]{y^{3}}}{\sqrt[3]{x^{2}}}}$

Step 1.1.3

Rewrite $x^{\frac{16}{20}}$ as $\sqrt[20]{x^{16}}$ .

$\sqrt{\frac{\sqrt[20]{x^{16}} \sqrt[4]{y^{3}}}{\sqrt[3]{x^{2}}}}$

Step 1.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{y^{3}}$ as $y^{\frac{3}{4}}$ .

$\sqrt{\frac{\sqrt[20]{x^{16}} y^{\frac{3}{4}}}{\sqrt[3]{x^{2}}}}$

Step 1.1.5

Rewrite $y^{\frac{3}{4}}$ as $y^{\frac{15}{20}}$ .

$\sqrt{\frac{\sqrt[20]{x^{16}} y^{\frac{15}{20}}}{\sqrt[3]{x^{2}}}}$

Step 1.1.6

Rewrite $y^{\frac{15}{20}}$ as $\sqrt[20]{y^{15}}$ .

$\sqrt{\frac{\sqrt[20]{x^{16}} \sqrt[20]{y^{15}}}{\sqrt[3]{x^{2}}}}$

Step 1.2

Combine using the product rule for radicals.

$\sqrt{\frac{\sqrt[20]{x^{16} y^{15}}}{\sqrt[3]{x^{2}}}}$

Step 2

Multiply $\frac{\sqrt[20]{x^{16} y^{15}}}{\sqrt[3]{x^{2}}}$ by $\frac{{\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{2}}$ .

$\sqrt{\frac{\sqrt[20]{x^{16} y^{15}}}{\sqrt[3]{x^{2}}} \cdot \frac{{\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{2}}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{\sqrt[20]{x^{16} y^{15}}}{\sqrt[3]{x^{2}}}$ by $\frac{{\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{2}}$ .

$\sqrt{\frac{\sqrt[20]{x^{16} y^{15}} {\sqrt[3]{x^{2}}}^{2}}{\sqrt[3]{x^{2}} {\sqrt[3]{x^{2}}}^{2}}}$

Step 3.2

Raise $\sqrt[3]{x^{2}}$ to the power of $1$ .

$\sqrt{\frac{\sqrt[20]{x^{16} y^{15}} {\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{1} {\sqrt[3]{x^{2}}}^{2}}}$

Step 3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sqrt{\frac{\sqrt[20]{x^{16} y^{15}} {\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{1 + 2}}}$

Step 3.4

Add $1$ and $2$ .

$\sqrt{\frac{\sqrt[20]{x^{16} y^{15}} {\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{3}}}$

Step 3.5

Rewrite ${\sqrt[3]{x^{2}}}^{3}$ as $x^{2}$ .

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Step 3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

$\sqrt{\frac{\sqrt[20]{x^{16} y^{15}} {\sqrt[3]{x^{2}}}^{2}}{{(x^{\frac{2}{3}})}^{3}}}$

Step 3.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{\frac{\sqrt[20]{x^{16} y^{15}} {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{2}{3} \cdot 3}}}$

Step 3.5.3

Combine $\frac{2}{3}$ and $3$ .

$\sqrt{\frac{\sqrt[20]{x^{16} y^{15}} {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{2 \cdot 3}{3}}}}$

Step 3.5.4

Multiply $2$ by $3$ .

$\sqrt{\frac{\sqrt[20]{x^{16} y^{15}} {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{6}{3}}}}$

Step 3.5.5

Cancel the common factor of $6$ and $3$ .

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Step 3.5.5.1

Factor $3$ out of $6$ .

$\sqrt{\frac{\sqrt[20]{x^{16} y^{15}} {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{3 \cdot 2}{3}}}}$

Step 3.5.5.2

Cancel the common factors.

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Step 3.5.5.2.1

Factor $3$ out of $3$ .

$\sqrt{\frac{\sqrt[20]{x^{16} y^{15}} {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{3 \cdot 2}{3 (1)}}}}$

Step 3.5.5.2.2

Cancel the common factor.

$\sqrt{\frac{\sqrt[20]{x^{16} y^{15}} {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{3 \cdot 2}{3 \cdot 1}}}}$

Step 3.5.5.2.3

Rewrite the expression.

$\sqrt{\frac{\sqrt[20]{x^{16} y^{15}} {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{2}{1}}}}$

Step 3.5.5.2.4

Divide $2$ by $1$ .

$\sqrt{\frac{\sqrt[20]{x^{16} y^{15}} {\sqrt[3]{x^{2}}}^{2}}{x^{2}}}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite ${\sqrt[3]{x^{2}}}^{2}$ as $\sqrt[3]{{(x^{2})}^{2}}$ .

$\sqrt{\frac{\sqrt[20]{x^{16} y^{15}} \sqrt[3]{{(x^{2})}^{2}}}{x^{2}}}$

Step 4.2

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 4.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{\frac{\sqrt[20]{x^{16} y^{15}} \sqrt[3]{x^{2 \cdot 2}}}{x^{2}}}$

Step 4.2.2

Multiply $2$ by $2$ .

$\sqrt{\frac{\sqrt[20]{x^{16} y^{15}} \sqrt[3]{x^{4}}}{x^{2}}}$

Step 4.3

Factor out $x^{3}$ .

$\sqrt{\frac{\sqrt[20]{x^{16} y^{15}} \sqrt[3]{x^{3} x}}{x^{2}}}$

Step 4.4

Pull terms out from under the radical.

$\sqrt{\frac{\sqrt[20]{x^{16} y^{15}} x \sqrt[3]{x}}{x^{2}}}$

Step 4.5

Combine exponents.

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Step 4.5.1

Rewrite the expression using the least common index of $60$ .

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Step 4.5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[20]{x^{16} y^{15}}$ as ${(x^{16} y^{15})}^{\frac{1}{20}}$ .

$\sqrt{\frac{x ({(x^{16} y^{15})}^{\frac{1}{20}} \sqrt[3]{x})}{x^{2}}}$

Step 4.5.1.2

Rewrite ${(x^{16} y^{15})}^{\frac{1}{20}}$ as ${(x^{16} y^{15})}^{\frac{3}{60}}$ .

$\sqrt{\frac{x ({(x^{16} y^{15})}^{\frac{3}{60}} \sqrt[3]{x})}{x^{2}}}$

Step 4.5.1.3

Rewrite ${(x^{16} y^{15})}^{\frac{3}{60}}$ as $\sqrt[60]{{(x^{16} y^{15})}^{3}}$ .

$\sqrt{\frac{x (\sqrt[60]{{(x^{16} y^{15})}^{3}} \sqrt[3]{x})}{x^{2}}}$

Step 4.5.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$\sqrt{\frac{x (\sqrt[60]{{(x^{16} y^{15})}^{3}} x^{\frac{1}{3}})}{x^{2}}}$

Step 4.5.1.5

Rewrite $x^{\frac{1}{3}}$ as $x^{\frac{20}{60}}$ .

$\sqrt{\frac{x (\sqrt[60]{{(x^{16} y^{15})}^{3}} x^{\frac{20}{60}})}{x^{2}}}$

Step 4.5.1.6

Rewrite $x^{\frac{20}{60}}$ as $\sqrt[60]{x^{20}}$ .

$\sqrt{\frac{x (\sqrt[60]{{(x^{16} y^{15})}^{3}} \sqrt[60]{x^{20}})}{x^{2}}}$

Step 4.5.2

Combine using the product rule for radicals.

$\sqrt{\frac{x \sqrt[60]{{(x^{16} y^{15})}^{3} x^{20}}}{x^{2}}}$

Step 4.5.3

Apply the product rule to $x^{16} y^{15}$ .

$\sqrt{\frac{x \sqrt[60]{{(x^{16})}^{3} {(y^{15})}^{3} x^{20}}}{x^{2}}}$

Step 4.5.4

Multiply the exponents in ${(x^{16})}^{3}$ .

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Step 4.5.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{\frac{x \sqrt[60]{x^{16 \cdot 3} {(y^{15})}^{3} x^{20}}}{x^{2}}}$

Step 4.5.4.2

Multiply $16$ by $3$ .

$\sqrt{\frac{x \sqrt[60]{x^{48} {(y^{15})}^{3} x^{20}}}{x^{2}}}$

Step 4.5.5

Multiply the exponents in ${(y^{15})}^{3}$ .

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Step 4.5.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{\frac{x \sqrt[60]{x^{48} y^{15 \cdot 3} x^{20}}}{x^{2}}}$

Step 4.5.5.2

Multiply $15$ by $3$ .

$\sqrt{\frac{x \sqrt[60]{x^{48} y^{45} x^{20}}}{x^{2}}}$

Step 4.5.6

Multiply $x^{48}$ by $x^{20}$ by adding the exponents.

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Step 4.5.6.1

Move $x^{20}$ .

$\sqrt{\frac{x \sqrt[60]{x^{20} x^{48} y^{45}}}{x^{2}}}$

Step 4.5.6.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sqrt{\frac{x \sqrt[60]{x^{20 + 48} y^{45}}}{x^{2}}}$

Step 4.5.6.3

Add $20$ and $48$ .

$\sqrt{\frac{x \sqrt[60]{x^{68} y^{45}}}{x^{2}}}$

Step 4.6

Rewrite $x^{68} y^{45}$ as $x^{60} (x^{8} y^{45})$ .

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Step 4.6.1

Factor out $x^{60}$ .

$\sqrt{\frac{x \sqrt[60]{x^{60} x^{8} y^{45}}}{x^{2}}}$

Step 4.6.2

Add parentheses.

$\sqrt{\frac{x \sqrt[60]{x^{60} (x^{8} y^{45})}}{x^{2}}}$

Step 4.7

Pull terms out from under the radical.

$\sqrt{\frac{x (x \sqrt[60]{x^{8} y^{45}})}{x^{2}}}$

Step 5

Simplify the numerator.

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Step 5.1

Raise $x$ to the power of $1$ .

$\sqrt{\frac{x^{1} x \sqrt[60]{x^{8} y^{45}}}{x^{2}}}$

Step 5.2

Raise $x$ to the power of $1$ .

$\sqrt{\frac{x^{1} x^{1} \sqrt[60]{x^{8} y^{45}}}{x^{2}}}$

Step 5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sqrt{\frac{x^{1 + 1} \sqrt[60]{x^{8} y^{45}}}{x^{2}}}$

Step 5.4

Add $1$ and $1$ .

$\sqrt{\frac{x^{2} \sqrt[60]{x^{8} y^{45}}}{x^{2}}}$

Step 6

Cancel the common factor of $x^{2}$ .

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Step 6.1

Cancel the common factor.

$\sqrt{\frac{x^{2} \sqrt[60]{x^{8} y^{45}}}{x^{2}}}$

Step 6.2

Divide $\sqrt[60]{x^{8} y^{45}}$ by $1$ .

$\sqrt{\sqrt[60]{x^{8} y^{45}}}$

Step 7

Rewrite $\sqrt{\sqrt[60]{x^{8} y^{45}}}$ as $\sqrt[120]{x^{8} y^{45}}$ .

$\sqrt[120]{x^{8} y^{45}}$