Simplify cube root of (18n^2)/(24n)

$\sqrt[3]{\frac{18 n^{2}}{24 n}}$

Step 1

Reduce the expression $\frac{18 n^{2}}{24 n}$ by cancelling the common factors.

Factor $6$ out of $18 n^{2}$ .

$\sqrt[3]{\frac{6 (3 n^{2})}{24 n}}$

Factor $6$ out of $24 n$ .

$\sqrt[3]{\frac{6 (3 n^{2})}{6 (4 n)}}$

Cancel the common factor.

$\sqrt[3]{\frac{6 (3 n^{2})}{6 (4 n)}}$

Rewrite the expression.

$\sqrt[3]{\frac{3 n^{2}}{4 n}}$

Step 2

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

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

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

Cancel the common factors.

Factor $n$ out of $4 n$ .

$\sqrt[3]{\frac{n (3 n)}{n \cdot 4}}$

Cancel the common factor.

$\sqrt[3]{\frac{n (3 n)}{n \cdot 4}}$

Rewrite the expression.

$\sqrt[3]{\frac{3 n}{4}}$

Step 3

Rewrite $\sqrt[3]{\frac{3 n}{4}}$ as $\frac{\sqrt[3]{3 n}}{\sqrt[3]{4}}$ .

$\frac{\sqrt[3]{3 n}}{\sqrt[3]{4}}$

Step 4

Multiply $\frac{\sqrt[3]{3 n}}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$\frac{\sqrt[3]{3 n}}{\sqrt[3]{4}} \cdot \frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$

Step 5

Combine and simplify the denominator.

Multiply $\frac{\sqrt[3]{3 n}}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$\frac{\sqrt[3]{3 n} {\sqrt[3]{4}}^{2}}{\sqrt[3]{4} {\sqrt[3]{4}}^{2}}$

Raise $\sqrt[3]{4}$ to the power of $1$ .

$\frac{\sqrt[3]{3 n} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1} {\sqrt[3]{4}}^{2}}$

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

$\frac{\sqrt[3]{3 n} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1 + 2}}$

Add $1$ and $2$ .

$\frac{\sqrt[3]{3 n} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{3}}$

Rewrite ${\sqrt[3]{4}}^{3}$ as $4$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4}$ as $4^{\frac{1}{3}}$ .

$\frac{\sqrt[3]{3 n} {\sqrt[3]{4}}^{2}}{{(4^{\frac{1}{3}})}^{3}}$

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

$\frac{\sqrt[3]{3 n} {\sqrt[3]{4}}^{2}}{4^{\frac{1}{3} \cdot 3}}$

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

$\frac{\sqrt[3]{3 n} {\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{\sqrt[3]{3 n} {\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}}$

Rewrite the expression.

$\frac{\sqrt[3]{3 n} {\sqrt[3]{4}}^{2}}{4^{1}}$

Evaluate the exponent.

$\frac{\sqrt[3]{3 n} {\sqrt[3]{4}}^{2}}{4}$

Step 6

Simplify the numerator.

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

$\frac{\sqrt[3]{3 n} \sqrt[3]{4^{2}}}{4}$

Raise $4$ to the power of $2$ .

$\frac{\sqrt[3]{3 n} \sqrt[3]{16}}{4}$

Rewrite $16$ as $2^{3} \cdot 2$ .

Factor $8$ out of $16$ .

$\frac{\sqrt[3]{3 n} \sqrt[3]{8 (2)}}{4}$

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

$\frac{\sqrt[3]{3 n} \sqrt[3]{2^{3} \cdot 2}}{4}$

Pull terms out from under the radical.

$\frac{\sqrt[3]{3 n} \cdot 2 \sqrt[3]{2}}{4}$

Combine exponents.

Combine using the product rule for radicals.

$\frac{2 \sqrt[3]{3 n \cdot 2}}{4}$

Multiply $2$ by $3$ .

$\frac{2 \sqrt[3]{6 n}}{4}$

Step 7

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2 \sqrt[3]{6 n}$ .

$\frac{2 (\sqrt[3]{6 n})}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$\frac{2 \sqrt[3]{6 n}}{2 \cdot 2}$

Cancel the common factor.

$\frac{2 \sqrt[3]{6 n}}{2 \cdot 2}$

Rewrite the expression.

$\frac{\sqrt[3]{6 n}}{2}$

Simplify cube root of (18n^2)/(24n)

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