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

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

Step 1

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

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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

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

Raise $\sqrt[3]{4 b^{2} c}$ to the power of $1$ .

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

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

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

Add $1$ and $2$ .

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

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 b^{2} c}$ as ${(4 b^{2} c)}^{\frac{1}{3}}$ .

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 4

Simplify the numerator.

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

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

Apply the product rule to $4 b^{2} c$ .

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

Apply the product rule to $4 b^{2}$ .

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

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

$\frac{\sqrt[3]{3 a} \sqrt[3]{16 {(b^{2})}^{2} c^{2}}}{4 b^{2} c}$

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

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

$\frac{\sqrt[3]{3 a} \sqrt[3]{16 b^{2 \cdot 2} c^{2}}}{4 b^{2} c}$

Multiply $2$ by $2$ .

$\frac{\sqrt[3]{3 a} \sqrt[3]{16 b^{4} c^{2}}}{4 b^{2} c}$

Rewrite $16 b^{4} c^{2}$ as ${(2 b)}^{3} \cdot (2 b c^{2})$ .

Factor $8$ out of $16$ .

$\frac{\sqrt[3]{3 a} \sqrt[3]{8 (2) b^{4} c^{2}}}{4 b^{2} c}$

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

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

Factor out $b^{3}$ .

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

Move $2$ .

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

Rewrite $2^{3} b^{3}$ as ${(2 b)}^{3}$ .

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

Add parentheses.

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

Add parentheses.

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

Pull terms out from under the radical.

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

Combine exponents.

Combine using the product rule for radicals.

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

Multiply $2$ by $3$ .

$\frac{2 b \sqrt[3]{6 a b c^{2}}}{4 b^{2} c}$

Step 5

Reduce the expression by cancelling the common factors.

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

Factor $2$ out of $2 b \sqrt[3]{6 a b c^{2}}$ .

$\frac{2 (b \sqrt[3]{6 a b c^{2}})}{4 b^{2} c}$

Cancel the common factors.

Factor $2$ out of $4 b^{2} c$ .

$\frac{2 (b \sqrt[3]{6 a b c^{2}})}{2 (2 b^{2} c)}$

Cancel the common factor.

$\frac{2 (b \sqrt[3]{6 a b c^{2}})}{2 (2 b^{2} c)}$

Rewrite the expression.

$\frac{b \sqrt[3]{6 a b c^{2}}}{2 b^{2} c}$

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

Factor $b$ out of $b \sqrt[3]{6 a b c^{2}}$ .

$\frac{b (\sqrt[3]{6 a b c^{2}})}{2 b^{2} c}$

Cancel the common factors.

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

$\frac{b (\sqrt[3]{6 a b c^{2}})}{b (2 b c)}$

Cancel the common factor.

$\frac{b \sqrt[3]{6 a b c^{2}}}{b (2 b c)}$

Rewrite the expression.

$\frac{\sqrt[3]{6 a b c^{2}}}{2 b c}$

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

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