Simplify the Radical Expression fourth root of 5/(x^3y^8)

$\sqrt[4]{\frac{5}{x^{3} y^{8}}}$

Rewrite $\frac{5}{x^{3} y^{8}}$ as ${(\frac{1}{y^{2}})}^{4} \frac{5}{x^{3}}$ .

Factor the perfect power $1^{4}$ out of $5$ .

$\sqrt[4]{\frac{1^{4} \cdot 5}{x^{3} y^{8}}}$

Factor the perfect power ${(y^{2})}^{4}$ out of $x^{3} y^{8}$ .

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

Rearrange the fraction $\frac{1^{4} \cdot 5}{{(y^{2})}^{4} x^{3}}$ .

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

Pull terms out from under the radical.

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

Remove non-negative terms from the absolute value.

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

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

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

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

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

Combine.

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

Multiply $\sqrt[4]{5}$ by $1$ .

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

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

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

Combine and simplify the denominator.

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

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

Move $\sqrt[4]{x^{3}}$ .

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

Raise $\sqrt[4]{x^{3}}$ to the power of $1$ .

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

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

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

Add $1$ and $3$ .

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

Rewrite ${\sqrt[4]{x^{3}}}^{4}$ as $x^{3}$ .

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

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

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

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

Combine $\frac{3}{4}$ and $4$ .

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

Multiply $3$ by $4$ .

$\frac{\sqrt[4]{5} {\sqrt[4]{x^{3}}}^{3}}{y^{2} x^{\frac{12}{4}}}$

Cancel the common factor of $12$ and $4$ .

Factor $4$ out of $12$ .

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

Cancel the common factors.

Factor $4$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $3$ by $1$ .

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

Simplify the numerator.

Rewrite ${\sqrt[4]{x^{3}}}^{3}$ as ${({(x^{3})}^{3})}^{\frac{1}{4}}$ .

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

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

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

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

Multiply $3$ by $3$ .

$\frac{\sqrt[4]{5} \sqrt[4]{x^{9}}}{y^{2} x^{3}}$

Rewrite $x^{9}$ as ${(x^{2})}^{4} x$ .

Factor out $x^{8}$ .

$\frac{\sqrt[4]{5} \sqrt[4]{x^{8} x}}{y^{2} x^{3}}$

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

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

Pull terms out from under the radical.

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

Remove non-negative terms from the absolute value.

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

Combine using the product rule for radicals.

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

Cancel the common factor of $x^{2}$ and $x^{3}$ .

Factor $x^{2}$ out of $x^{2} \sqrt[4]{5 x}$ .

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

Cancel the common factors.

Factor $x^{2}$ out of $y^{2} x^{3}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the Radical Expression fourth root of 5/(x^3y^8)

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