Algebra Examples

$\sqrt[4]{\frac{32 x^{6}}{y^{2}}}$

Step 1

Rewrite $\sqrt[4]{\frac{32 x^{6}}{y^{2}}}$ as $\frac{\sqrt[4]{32 x^{6}}}{\sqrt[4]{y^{2}}}$ .

$\frac{\sqrt[4]{32 x^{6}}}{\sqrt[4]{y^{2}}}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $32 x^{6}$ as ${(2 x)}^{4} \cdot (2 x^{2})$ .

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Step 2.1.1

Factor $16$ out of $32$ .

$\frac{\sqrt[4]{16 (2) x^{6}}}{\sqrt[4]{y^{2}}}$

Step 2.1.2

Rewrite $16$ as $2^{4}$ .

$\frac{\sqrt[4]{2^{4} \cdot 2 x^{6}}}{\sqrt[4]{y^{2}}}$

Step 2.1.3

Factor out $x^{4}$ .

$\frac{\sqrt[4]{2^{4} \cdot 2 (x^{4} x^{2})}}{\sqrt[4]{y^{2}}}$

Step 2.1.4

Move $2$ .

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

Step 2.1.5

Rewrite $2^{4} x^{4}$ as ${(2 x)}^{4}$ .

$\frac{\sqrt[4]{{(2 x)}^{4} \cdot 2 x^{2}}}{\sqrt[4]{y^{2}}}$

Step 2.1.6

Add parentheses.

$\frac{\sqrt[4]{{(2 x)}^{4} \cdot (2 x^{2})}}{\sqrt[4]{y^{2}}}$

Step 2.2

Pull terms out from under the radical.

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

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $\sqrt[4]{y^{2}}$ as $\sqrt{\sqrt{y^{2}}}$ .

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

Step 3.2

Pull terms out from under the radical, assuming positive real numbers.

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

Step 4

Multiply $\frac{2 x \sqrt[4]{2 x^{2}}}{\sqrt{y}}$ by $\frac{\sqrt{y}}{\sqrt{y}}$ .

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

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{2 x \sqrt[4]{2 x^{2}}}{\sqrt{y}}$ by $\frac{\sqrt{y}}{\sqrt{y}}$ .

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

Step 5.2

Raise $\sqrt{y}$ to the power of $1$ .

$\frac{2 x \sqrt[4]{2 x^{2}} \sqrt{y}}{{\sqrt{y}}^{1} \sqrt{y}}$

Step 5.3

Raise $\sqrt{y}$ to the power of $1$ .

$\frac{2 x \sqrt[4]{2 x^{2}} \sqrt{y}}{{\sqrt{y}}^{1} {\sqrt{y}}^{1}}$

Step 5.4

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

$\frac{2 x \sqrt[4]{2 x^{2}} \sqrt{y}}{{\sqrt{y}}^{1 + 1}}$

Step 5.5

Add $1$ and $1$ .

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

Step 5.6

Rewrite ${\sqrt{y}}^{2}$ as $y$ .

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Step 5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

$\frac{2 x \sqrt[4]{2 x^{2}} \sqrt{y}}{{(y^{\frac{1}{2}})}^{2}}$

Step 5.6.2

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

$\frac{2 x \sqrt[4]{2 x^{2}} \sqrt{y}}{y^{\frac{1}{2} \cdot 2}}$

Step 5.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 5.6.4

Cancel the common factor of $2$ .

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Step 5.6.4.1

Cancel the common factor.

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

Step 5.6.4.2

Rewrite the expression.

$\frac{2 x \sqrt[4]{2 x^{2}} \sqrt{y}}{y^{1}}$

Step 5.6.5

Simplify.

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

Step 6

Simplify the numerator.

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Step 6.1

Rewrite the expression using the least common index of $4$ .

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Step 6.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

$\frac{2 x (y^{\frac{1}{2}} \sqrt[4]{2 x^{2}})}{y}$

Step 6.1.2

Rewrite $y^{\frac{1}{2}}$ as $y^{\frac{2}{4}}$ .

$\frac{2 x (y^{\frac{2}{4}} \sqrt[4]{2 x^{2}})}{y}$

Step 6.1.3

Rewrite $y^{\frac{2}{4}}$ as $\sqrt[4]{y^{2}}$ .

$\frac{2 x (\sqrt[4]{y^{2}} \sqrt[4]{2 x^{2}})}{y}$

Step 6.2

Combine using the product rule for radicals.

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

Step 7

Move $2$ to the left of $y^{2}$ .

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