Simplify ( cube root of 32)/( cube root of 12)

$\frac{\sqrt[3]{32}}{\sqrt[3]{12}}$

Step 1

Combine $\sqrt[3]{32}$ and $\sqrt[3]{12}$ into a single radical.

$\sqrt[3]{\frac{32}{12}}$

Step 2

Cancel the common factor of $32$ and $12$ .

Factor $4$ out of $32$ .

$\sqrt[3]{\frac{4 (8)}{12}}$

Cancel the common factors.

Factor $4$ out of $12$ .

$\sqrt[3]{\frac{4 \cdot 8}{4 \cdot 3}}$

Cancel the common factor.

$\sqrt[3]{\frac{4 \cdot 8}{4 \cdot 3}}$

Rewrite the expression.

$\sqrt[3]{\frac{8}{3}}$

Step 3

Rewrite $\sqrt[3]{\frac{8}{3}}$ as $\frac{\sqrt[3]{8}}{\sqrt[3]{3}}$ .

$\frac{\sqrt[3]{8}}{\sqrt[3]{3}}$

Step 4

Simplify the numerator.

Rewrite $8$ as $2^{3}$ .

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

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

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

Step 5

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

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

Step 6

Combine and simplify the denominator.

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

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

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

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

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

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

Add $1$ and $2$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 7

Simplify the numerator.

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

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

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

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.38672254 \dots$

Simplify ( cube root of 32)/( cube root of 12)

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