Evaluate cube root of (2^15)/(3^2)

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

Step 1

Raise $2$ to the power of $15$ .

$\sqrt[3]{\frac{32768}{3^{2}}}$

Step 2

Raise $3$ to the power of $2$ .

$\sqrt[3]{\frac{32768}{9}}$

Step 3

Rewrite $\sqrt[3]{\frac{32768}{9}}$ as $\frac{\sqrt[3]{32768}}{\sqrt[3]{9}}$ .

$\frac{\sqrt[3]{32768}}{\sqrt[3]{9}}$

Step 4

Simplify the numerator.

Rewrite $32768$ as $32^{3}$ .

$\frac{\sqrt[3]{32^{3}}}{\sqrt[3]{9}}$

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

$\frac{32}{\sqrt[3]{9}}$

Step 5

Multiply $\frac{32}{\sqrt[3]{9}}$ by $\frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{2}}$ .

$\frac{32}{\sqrt[3]{9}} \cdot \frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{2}}$

Step 6

Combine and simplify the denominator.

Multiply $\frac{32}{\sqrt[3]{9}}$ by $\frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{2}}$ .

$\frac{32 {\sqrt[3]{9}}^{2}}{\sqrt[3]{9} {\sqrt[3]{9}}^{2}}$

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

$\frac{32 {\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{1} {\sqrt[3]{9}}^{2}}$

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

$\frac{32 {\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{1 + 2}}$

Add $1$ and $2$ .

$\frac{32 {\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{3}}$

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

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

$\frac{32 {\sqrt[3]{9}}^{2}}{{(9^{\frac{1}{3}})}^{3}}$

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

$\frac{32 {\sqrt[3]{9}}^{2}}{9^{\frac{1}{3} \cdot 3}}$

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

$\frac{32 {\sqrt[3]{9}}^{2}}{9^{\frac{3}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{32 {\sqrt[3]{9}}^{2}}{9^{\frac{3}{3}}}$

Rewrite the expression.

$\frac{32 {\sqrt[3]{9}}^{2}}{9^{1}}$

Evaluate the exponent.

$\frac{32 {\sqrt[3]{9}}^{2}}{9}$

Step 7

Simplify the numerator.

Rewrite ${\sqrt[3]{9}}^{2}$ as ${(9^{2})}^{\frac{1}{3}}$ .

$\frac{32 \sqrt[3]{9^{2}}}{9}$

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

$\frac{32 \sqrt[3]{81}}{9}$

Rewrite $81$ as $3^{3} \cdot 3$ .

Factor $27$ out of $81$ .

$\frac{32 \sqrt[3]{27 (3)}}{9}$

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

$\frac{32 \sqrt[3]{3^{3} \cdot 3}}{9}$

Pull terms out from under the radical.

$\frac{32 \cdot 3 \sqrt[3]{3}}{9}$

Multiply $32$ by $3$ .

$\frac{96 \sqrt[3]{3}}{9}$

Step 8

Cancel the common factor of $96$ and $9$ .

Factor $3$ out of $96 \sqrt[3]{3}$ .

$\frac{3 (32 \sqrt[3]{3})}{9}$

Cancel the common factors.

Factor $3$ out of $9$ .

$\frac{3 (32 \sqrt[3]{3})}{3 (3)}$

Cancel the common factor.

$\frac{3 (32 \sqrt[3]{3})}{3 \cdot 3}$

Rewrite the expression.

$\frac{32 \sqrt[3]{3}}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{32 \sqrt[3]{3}}{3}$

Decimal Form:

$15.38399541 \dots$

Evaluate cube root of (2^15)/(3^2)

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