Simplify cube root of 5/(9x^2)

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

Step 1

Rewrite $\sqrt[3]{\frac{5}{9 x^{2}}}$ as $\frac{\sqrt[3]{5}}{\sqrt[3]{9 x^{2}}}$ .

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

Step 2

Multiply $\frac{\sqrt[3]{5}}{\sqrt[3]{9 x^{2}}}$ by $\frac{{\sqrt[3]{9 x^{2}}}^{2}}{{\sqrt[3]{9 x^{2}}}^{2}}$ .

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

Step 3

Combine and simplify the denominator.

Multiply $\frac{\sqrt[3]{5}}{\sqrt[3]{9 x^{2}}}$ by $\frac{{\sqrt[3]{9 x^{2}}}^{2}}{{\sqrt[3]{9 x^{2}}}^{2}}$ .

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

Raise $\sqrt[3]{9 x^{2}}$ to the power of $1$ .

$\frac{\sqrt[3]{5} {\sqrt[3]{9 x^{2}}}^{2}}{{\sqrt[3]{9 x^{2}}}^{1} {\sqrt[3]{9 x^{2}}}^{2}}$

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

$\frac{\sqrt[3]{5} {\sqrt[3]{9 x^{2}}}^{2}}{{\sqrt[3]{9 x^{2}}}^{1 + 2}}$

Add $1$ and $2$ .

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

Rewrite ${\sqrt[3]{9 x^{2}}}^{3}$ as $9 x^{2}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{9 x^{2}}$ as ${(9 x^{2})}^{\frac{1}{3}}$ .

$\frac{\sqrt[3]{5} {\sqrt[3]{9 x^{2}}}^{2}}{{({(9 x^{2})}^{\frac{1}{3}})}^{3}}$

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

$\frac{\sqrt[3]{5} {\sqrt[3]{9 x^{2}}}^{2}}{{(9 x^{2})}^{\frac{1}{3} \cdot 3}}$

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

$\frac{\sqrt[3]{5} {\sqrt[3]{9 x^{2}}}^{2}}{{(9 x^{2})}^{\frac{3}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{\sqrt[3]{5} {\sqrt[3]{9 x^{2}}}^{2}}{{(9 x^{2})}^{\frac{3}{3}}}$

Rewrite the expression.

$\frac{\sqrt[3]{5} {\sqrt[3]{9 x^{2}}}^{2}}{{(9 x^{2})}^{1}}$

Simplify.

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

Step 4

Simplify the numerator.

Rewrite ${\sqrt[3]{9 x^{2}}}^{2}$ as $\sqrt[3]{{(9 x^{2})}^{2}}$ .

$\frac{\sqrt[3]{5} \sqrt[3]{{(9 x^{2})}^{2}}}{9 x^{2}}$

Apply the product rule to $9 x^{2}$ .

$\frac{\sqrt[3]{5} \sqrt[3]{9^{2} {(x^{2})}^{2}}}{9 x^{2}}$

Raise $9$ to the power of $2$ .

$\frac{\sqrt[3]{5} \sqrt[3]{81 {(x^{2})}^{2}}}{9 x^{2}}$

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

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

$\frac{\sqrt[3]{5} \sqrt[3]{81 x^{2 \cdot 2}}}{9 x^{2}}$

Multiply $2$ by $2$ .

$\frac{\sqrt[3]{5} \sqrt[3]{81 x^{4}}}{9 x^{2}}$

Rewrite $81 x^{4}$ as ${(3 x)}^{3} \cdot (3 x)$ .

Factor $27$ out of $81$ .

$\frac{\sqrt[3]{5} \sqrt[3]{27 (3) x^{4}}}{9 x^{2}}$

Rewrite $27$ as $3^{3}$ .

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

Factor out $x^{3}$ .

$\frac{\sqrt[3]{5} \sqrt[3]{3^{3} \cdot 3 (x^{3} x)}}{9 x^{2}}$

Move $3$ .

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

Rewrite $3^{3} x^{3}$ as ${(3 x)}^{3}$ .

$\frac{\sqrt[3]{5} \sqrt[3]{{(3 x)}^{3} \cdot 3 x}}{9 x^{2}}$

Add parentheses.

$\frac{\sqrt[3]{5} \sqrt[3]{{(3 x)}^{3} \cdot (3 x)}}{9 x^{2}}$

Pull terms out from under the radical.

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

Combine exponents.

Combine using the product rule for radicals.

$\frac{3 x \sqrt[3]{5 (3 x)}}{9 x^{2}}$

Multiply $5$ by $3$ .

$\frac{3 x \sqrt[3]{15 x}}{9 x^{2}}$

Step 5

Reduce the expression by cancelling the common factors.

Cancel the common factor of $3$ and $9$ .

Factor $3$ out of $3 x \sqrt[3]{15 x}$ .

$\frac{3 (x \sqrt[3]{15 x})}{9 x^{2}}$

Cancel the common factors.

Factor $3$ out of $9 x^{2}$ .

$\frac{3 (x \sqrt[3]{15 x})}{3 (3 x^{2})}$

Cancel the common factor.

$\frac{3 (x \sqrt[3]{15 x})}{3 (3 x^{2})}$

Rewrite the expression.

$\frac{x \sqrt[3]{15 x}}{3 x^{2}}$

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

Factor $x$ out of $x \sqrt[3]{15 x}$ .

$\frac{x (\sqrt[3]{15 x})}{3 x^{2}}$

Cancel the common factors.

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

$\frac{x (\sqrt[3]{15 x})}{x (3 x)}$

Cancel the common factor.

$\frac{x \sqrt[3]{15 x}}{x (3 x)}$

Rewrite the expression.

$\frac{\sqrt[3]{15 x}}{3 x}$

Simplify cube root of 5/(9x^2)

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