Algebra Examples

$\sqrt[6]{\frac{16 x^{4}}{25 y^{2}}}$

Step 1

Rewrite $16 x^{4}$ as ${(4 x^{2})}^{2}$ .

$\sqrt[6]{\frac{{(4 x^{2})}^{2}}{25 y^{2}}}$

Step 2

Rewrite $25 y^{2}$ as ${(5 y)}^{2}$ .

$\sqrt[6]{\frac{{(4 x^{2})}^{2}}{{(5 y)}^{2}}}$

Step 3

Rewrite $\frac{{(4 x^{2})}^{2}}{{(5 y)}^{2}}$ as ${(\frac{4 x^{2}}{5 y})}^{2}$ .

$\sqrt[6]{{(\frac{4 x^{2}}{5 y})}^{2}}$

Step 4

Rewrite $\sqrt[6]{{(\frac{4 x^{2}}{5 y})}^{2}}$ as $\sqrt[3]{\sqrt{{(\frac{4 x^{2}}{5 y})}^{2}}}$ .

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Combine and simplify the denominator.

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Step 8.1

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

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

Step 8.2

Raise $\sqrt[3]{5 y}$ to the power of $1$ .

$\frac{\sqrt[3]{4 x^{2}} {\sqrt[3]{5 y}}^{2}}{{\sqrt[3]{5 y}}^{1} {\sqrt[3]{5 y}}^{2}}$

Step 8.3

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

$\frac{\sqrt[3]{4 x^{2}} {\sqrt[3]{5 y}}^{2}}{{\sqrt[3]{5 y}}^{1 + 2}}$

Step 8.4

Add $1$ and $2$ .

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

Step 8.5

Rewrite ${\sqrt[3]{5 y}}^{3}$ as $5 y$ .

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Step 8.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5 y}$ as ${(5 y)}^{\frac{1}{3}}$ .

$\frac{\sqrt[3]{4 x^{2}} {\sqrt[3]{5 y}}^{2}}{{({(5 y)}^{\frac{1}{3}})}^{3}}$

Step 8.5.2

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

$\frac{\sqrt[3]{4 x^{2}} {\sqrt[3]{5 y}}^{2}}{{(5 y)}^{\frac{1}{3} \cdot 3}}$

Step 8.5.3

Combine $\frac{1}{3}$ and $3$ .

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

Step 8.5.4

Cancel the common factor of $3$ .

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Step 8.5.4.1

Cancel the common factor.

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

Step 8.5.4.2

Rewrite the expression.

$\frac{\sqrt[3]{4 x^{2}} {\sqrt[3]{5 y}}^{2}}{{(5 y)}^{1}}$

Step 8.5.5

Simplify.

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

Step 9

Simplify the numerator.

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Step 9.1

Rewrite ${\sqrt[3]{5 y}}^{2}$ as $\sqrt[3]{{(5 y)}^{2}}$ .

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

Step 9.2

Apply the product rule to $5 y$ .

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

Step 9.3

Raise $5$ to the power of $2$ .

$\frac{\sqrt[3]{4 x^{2}} \sqrt[3]{25 y^{2}}}{5 y}$

Step 10

Simplify the numerator.

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Step 10.1

Combine using the product rule for radicals.

$\frac{\sqrt[3]{4 x^{2} (25 y^{2})}}{5 y}$

Step 10.2

Multiply $25$ by $4$ .

$\frac{\sqrt[3]{100 x^{2} y^{2}}}{5 y}$