Simplify 1/( cube root of 9x)

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

Step 1

Multiply $\frac{1}{\sqrt[3]{9 x}}$ by $\frac{{\sqrt[3]{9 x}}^{2}}{{\sqrt[3]{9 x}}^{2}}$ .

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

Step 2

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt[3]{9 x}}$ by $\frac{{\sqrt[3]{9 x}}^{2}}{{\sqrt[3]{9 x}}^{2}}$ .

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

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

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

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

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

Add $1$ and $2$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 3

Simplify the numerator.

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

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

Apply the product rule to $9 x$ .

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

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

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

Rewrite $81 x^{2}$ as $3^{3} \cdot (3 x^{2})$ .

Factor $27$ out of $81$ .

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

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

Step 4

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

Factor $3$ out of $3 \sqrt[3]{3 x^{2}}$ .

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

Cancel the common factors.

Factor $3$ out of $9 x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify 1/( cube root of 9x)

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