Simplify sixth root of (16x^4)/(25y^2)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

Add $1$ and $2$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the numerator.

Rewrite ${\sqrt[3]{5 y}}^{2}$ as ${({(5 y)}^{2})}^{\frac{1}{3}}$ .

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

Apply the product rule to $5 y$ .

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

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

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

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $25$ by $4$ .

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

Simplify sixth root of (16x^4)/(25y^2)

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