Simplify ( square root of x^4y)/( fourth root of x^2y^8)

$\frac{\sqrt{x^{4} y}}{\sqrt[4]{x^{2} y^{8}}}$

Simplify the numerator.

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

$\frac{\sqrt{{(x^{2})}^{2} y}}{\sqrt[4]{x^{2} y^{8}}}$

Pull terms out from under the radical.

$\frac{x^{2} \sqrt{y}}{\sqrt[4]{x^{2} y^{8}}}$

Simplify the denominator.

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Rewrite $x^{2} y^{8}$ as ${(y^{2})}^{4} x^{2}$ .

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Rewrite $y^{8}$ as ${(y^{2})}^{4}$ .

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

Reorder $x^{2}$ and ${(y^{2})}^{4}$ .

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

Pull terms out from under the radical.

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

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

$\frac{x^{2} \sqrt{y}}{y^{2} \sqrt{\sqrt{x^{2}}}}$

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

$\frac{x^{2} \sqrt{y}}{y^{2} \sqrt{x}}$

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

$\frac{x^{2} \sqrt{y}}{y^{2} \sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}$

Combine and simplify the denominator.

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Multiply $\frac{x^{2} \sqrt{y}}{y^{2} \sqrt{x}}$ and $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{x^{2} \sqrt{y} \sqrt{x}}{y^{2} \sqrt{x} \sqrt{x}}$

Move $\sqrt{x}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

$\frac{x^{2} \sqrt{y} \sqrt{x}}{y^{2} {\sqrt{x}}^{2}}$

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

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

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

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

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

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

$\frac{x^{2} \sqrt{y} \sqrt{x}}{y^{2} x^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{x^{2} \sqrt{y} \sqrt{x}}{y^{2} x^{\frac{2}{2}}}$

Divide $1$ by $1$ .

$\frac{x^{2} \sqrt{y} \sqrt{x}}{y^{2} x^{1}}$

Simplify.

$\frac{x^{2} \sqrt{y} \sqrt{x}}{y^{2} x}$

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

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Factor $x$ out of $x^{2} \sqrt{y} \sqrt{x}$ .

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

Cancel the common factors.

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Factor $x$ out of $y^{2} x$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{x \sqrt{y} \sqrt{x}}{y^{2}}$

Combine using the product rule for radicals.

$\frac{x \sqrt{x y}}{y^{2}}$

Simplify ( square root of x^4y)/( fourth root of x^2y^8)

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