Simplify fifth root of (10x)/(3x^3)

$\sqrt[5]{\frac{10 x}{3 x^{3}}}$

Step 1

Reduce the expression $\frac{10 x}{3 x^{3}}$ by cancelling the common factors.

Factor $x$ out of $10 x$ .

$\sqrt[5]{\frac{x \cdot 10}{3 x^{3}}}$

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

$\sqrt[5]{\frac{x \cdot 10}{x (3 x^{2})}}$

Cancel the common factor.

$\sqrt[5]{\frac{x \cdot 10}{x (3 x^{2})}}$

Rewrite the expression.

$\sqrt[5]{\frac{10}{3 x^{2}}}$

Step 2

Rewrite $\sqrt[5]{\frac{10}{3 x^{2}}}$ as $\frac{\sqrt[5]{10}}{\sqrt[5]{3 x^{2}}}$ .

$\frac{\sqrt[5]{10}}{\sqrt[5]{3 x^{2}}}$

Step 3

Multiply $\frac{\sqrt[5]{10}}{\sqrt[5]{3 x^{2}}}$ by $\frac{{\sqrt[5]{3 x^{2}}}^{4}}{{\sqrt[5]{3 x^{2}}}^{4}}$ .

$\frac{\sqrt[5]{10}}{\sqrt[5]{3 x^{2}}} \cdot \frac{{\sqrt[5]{3 x^{2}}}^{4}}{{\sqrt[5]{3 x^{2}}}^{4}}$

Step 4

Combine and simplify the denominator.

Multiply $\frac{\sqrt[5]{10}}{\sqrt[5]{3 x^{2}}}$ by $\frac{{\sqrt[5]{3 x^{2}}}^{4}}{{\sqrt[5]{3 x^{2}}}^{4}}$ .

$\frac{\sqrt[5]{10} {\sqrt[5]{3 x^{2}}}^{4}}{\sqrt[5]{3 x^{2}} {\sqrt[5]{3 x^{2}}}^{4}}$

Raise $\sqrt[5]{3 x^{2}}$ to the power of $1$ .

$\frac{\sqrt[5]{10} {\sqrt[5]{3 x^{2}}}^{4}}{{\sqrt[5]{3 x^{2}}}^{1} {\sqrt[5]{3 x^{2}}}^{4}}$

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

$\frac{\sqrt[5]{10} {\sqrt[5]{3 x^{2}}}^{4}}{{\sqrt[5]{3 x^{2}}}^{1 + 4}}$

Add $1$ and $4$ .

$\frac{\sqrt[5]{10} {\sqrt[5]{3 x^{2}}}^{4}}{{\sqrt[5]{3 x^{2}}}^{5}}$

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{3 x^{2}}$ as ${(3 x^{2})}^{\frac{1}{5}}$ .

$\frac{\sqrt[5]{10} {\sqrt[5]{3 x^{2}}}^{4}}{{({(3 x^{2})}^{\frac{1}{5}})}^{5}}$

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

$\frac{\sqrt[5]{10} {\sqrt[5]{3 x^{2}}}^{4}}{{(3 x^{2})}^{\frac{1}{5} \cdot 5}}$

Combine $\frac{1}{5}$ and $5$ .

$\frac{\sqrt[5]{10} {\sqrt[5]{3 x^{2}}}^{4}}{{(3 x^{2})}^{\frac{5}{5}}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{\sqrt[5]{10} {\sqrt[5]{3 x^{2}}}^{4}}{{(3 x^{2})}^{\frac{5}{5}}}$

Rewrite the expression.

$\frac{\sqrt[5]{10} {\sqrt[5]{3 x^{2}}}^{4}}{{(3 x^{2})}^{1}}$

Simplify.

$\frac{\sqrt[5]{10} {\sqrt[5]{3 x^{2}}}^{4}}{3 x^{2}}$

Step 5

Simplify the numerator.

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

$\frac{\sqrt[5]{10} \sqrt[5]{{(3 x^{2})}^{4}}}{3 x^{2}}$

Apply the product rule to $3 x^{2}$ .

$\frac{\sqrt[5]{10} \sqrt[5]{3^{4} {(x^{2})}^{4}}}{3 x^{2}}$

Raise $3$ to the power of $4$ .

$\frac{\sqrt[5]{10} \sqrt[5]{81 {(x^{2})}^{4}}}{3 x^{2}}$

Multiply the exponents in ${(x^{2})}^{4}$ .

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

$\frac{\sqrt[5]{10} \sqrt[5]{81 x^{2 \cdot 4}}}{3 x^{2}}$

Multiply $2$ by $4$ .

$\frac{\sqrt[5]{10} \sqrt[5]{81 x^{8}}}{3 x^{2}}$

Rewrite $81 x^{8}$ as $x^{5} \cdot (81 x^{3})$ .

Factor out $x^{5}$ .

$\frac{\sqrt[5]{10} \sqrt[5]{81 (x^{5} x^{3})}}{3 x^{2}}$

Reorder $81$ and $x^{5}$ .

$\frac{\sqrt[5]{10} \sqrt[5]{x^{5} \cdot 81 x^{3}}}{3 x^{2}}$

Add parentheses.

$\frac{\sqrt[5]{10} \sqrt[5]{x^{5} \cdot (81 x^{3})}}{3 x^{2}}$

Pull terms out from under the radical.

$\frac{\sqrt[5]{10} x \sqrt[5]{81 x^{3}}}{3 x^{2}}$

Combine exponents.

Combine using the product rule for radicals.

$\frac{x \sqrt[5]{10 (81 x^{3})}}{3 x^{2}}$

Multiply $10$ by $81$ .

$\frac{x \sqrt[5]{810 x^{3}}}{3 x^{2}}$

Step 6

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

Factor $x$ out of $x \sqrt[5]{810 x^{3}}$ .

$\frac{x (\sqrt[5]{810 x^{3}})}{3 x^{2}}$

Cancel the common factors.

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

$\frac{x (\sqrt[5]{810 x^{3}})}{x (3 x)}$

Cancel the common factor.

$\frac{x \sqrt[5]{810 x^{3}}}{x (3 x)}$

Rewrite the expression.

$\frac{\sqrt[5]{810 x^{3}}}{3 x}$

Simplify fifth root of (10x)/(3x^3)

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