Simplify cube root of (4a^3)/(81b^2)

$\sqrt[3]{\frac{4 a^{3}}{81 b^{2}}}$

Step 1

Rewrite $\frac{4 a^{3}}{81 b^{2}}$ as ${(\frac{a}{3})}^{3} \frac{4}{3 b^{2}}$ .

Factor the perfect power $a^{3}$ out of $4 a^{3}$ .

$\sqrt[3]{\frac{a^{3} \cdot 4}{81 b^{2}}}$

Factor the perfect power $3^{3}$ out of $81 b^{2}$ .

$\sqrt[3]{\frac{a^{3} \cdot 4}{3^{3} \cdot (3 b^{2})}}$

Rearrange the fraction $\frac{a^{3} \cdot 4}{3^{3} \cdot (3 b^{2})}$ .

$\sqrt[3]{{(\frac{a}{3})}^{3} \frac{4}{3 b^{2}}}$

Step 2

Pull terms out from under the radical.

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

Step 3

Rewrite $\sqrt[3]{\frac{4}{3 b^{2}}}$ as $\frac{\sqrt[3]{4}}{\sqrt[3]{3 b^{2}}}$ .

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

Step 4

Combine.

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

Step 5

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

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

Step 6

Combine and simplify the denominator.

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

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

Move $\sqrt[3]{3 b^{2}}$ .

$\frac{a \sqrt[3]{4} {\sqrt[3]{3 b^{2}}}^{2}}{3 (\sqrt[3]{3 b^{2}} {\sqrt[3]{3 b^{2}}}^{2})}$

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

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

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

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

Add $1$ and $2$ .

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

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

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

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

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

$\frac{a \sqrt[3]{4} {\sqrt[3]{3 b^{2}}}^{2}}{3 {(3 b^{2})}^{\frac{1}{3} \cdot 3}}$

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

$\frac{a \sqrt[3]{4} {\sqrt[3]{3 b^{2}}}^{2}}{3 {(3 b^{2})}^{\frac{3}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{a \sqrt[3]{4} {\sqrt[3]{3 b^{2}}}^{2}}{3 {(3 b^{2})}^{\frac{3}{3}}}$

Rewrite the expression.

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

Simplify.

$\frac{a \sqrt[3]{4} {\sqrt[3]{3 b^{2}}}^{2}}{3 (3 b^{2})}$

Step 7

Simplify the numerator.

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

$\frac{a \sqrt[3]{4} \sqrt[3]{{(3 b^{2})}^{2}}}{3 \cdot 3 b^{2}}$

Apply the product rule to $3 b^{2}$ .

$\frac{a \sqrt[3]{4} \sqrt[3]{3^{2} {(b^{2})}^{2}}}{3 \cdot 3 b^{2}}$

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

$\frac{a \sqrt[3]{4} \sqrt[3]{9 {(b^{2})}^{2}}}{3 \cdot 3 b^{2}}$

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

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

$\frac{a \sqrt[3]{4} \sqrt[3]{9 b^{2 \cdot 2}}}{3 \cdot 3 b^{2}}$

Multiply $2$ by $2$ .

$\frac{a \sqrt[3]{4} \sqrt[3]{9 b^{4}}}{3 \cdot 3 b^{2}}$

Rewrite $9 b^{4}$ as $b^{3} \cdot (9 b)$ .

Factor out $b^{3}$ .

$\frac{a \sqrt[3]{4} \sqrt[3]{9 (b^{3} b)}}{3 \cdot 3 b^{2}}$

Reorder $9$ and $b^{3}$ .

$\frac{a \sqrt[3]{4} \sqrt[3]{b^{3} \cdot 9 b}}{3 \cdot 3 b^{2}}$

Add parentheses.

$\frac{a \sqrt[3]{4} \sqrt[3]{b^{3} \cdot (9 b)}}{3 \cdot 3 b^{2}}$

Pull terms out from under the radical.

$\frac{a \sqrt[3]{4} b \sqrt[3]{9 b}}{3 \cdot 3 b^{2}}$

Combine exponents.

Combine using the product rule for radicals.

$\frac{a b \sqrt[3]{4 (9 b)}}{3 \cdot 3 b^{2}}$

Multiply $4$ by $9$ .

$\frac{a b \sqrt[3]{36 b}}{3 \cdot 3 b^{2}}$

Step 8

Reduce the expression by cancelling the common factors.

Multiply $3$ by $3$ .

$\frac{a b \sqrt[3]{36 b}}{3^{2} b^{2}}$

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

$\frac{a b \sqrt[3]{36 b}}{9 b^{2}}$

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

Factor $b$ out of $a b \sqrt[3]{36 b}$ .

$\frac{b (a \sqrt[3]{36 b})}{9 b^{2}}$

Cancel the common factors.

Factor $b$ out of $9 b^{2}$ .

$\frac{b (a \sqrt[3]{36 b})}{b (9 b)}$

Cancel the common factor.

$\frac{b (a \sqrt[3]{36 b})}{b (9 b)}$

Rewrite the expression.

$\frac{a \sqrt[3]{36 b}}{9 b}$

Simplify cube root of (4a^3)/(81b^2)

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