Simplify the Radical Expression fourth root of (3a^2)/(8b^3)

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

Step 1

Rewrite $\sqrt[4]{\frac{3 a^{2}}{8 b^{3}}}$ as $\frac{\sqrt[4]{3 a^{2}}}{\sqrt[4]{8 b^{3}}}$ .

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

Step 2

Multiply $\frac{\sqrt[4]{3 a^{2}}}{\sqrt[4]{8 b^{3}}}$ by $\frac{{\sqrt[4]{8 b^{3}}}^{3}}{{\sqrt[4]{8 b^{3}}}^{3}}$ .

$\frac{\sqrt[4]{3 a^{2}}}{\sqrt[4]{8 b^{3}}} \cdot \frac{{\sqrt[4]{8 b^{3}}}^{3}}{{\sqrt[4]{8 b^{3}}}^{3}}$

Step 3

Combine and simplify the denominator.

Multiply $\frac{\sqrt[4]{3 a^{2}}}{\sqrt[4]{8 b^{3}}}$ by $\frac{{\sqrt[4]{8 b^{3}}}^{3}}{{\sqrt[4]{8 b^{3}}}^{3}}$ .

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

Raise $\sqrt[4]{8 b^{3}}$ to the power of $1$ .

$\frac{\sqrt[4]{3 a^{2}} {\sqrt[4]{8 b^{3}}}^{3}}{{\sqrt[4]{8 b^{3}}}^{1} {\sqrt[4]{8 b^{3}}}^{3}}$

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

$\frac{\sqrt[4]{3 a^{2}} {\sqrt[4]{8 b^{3}}}^{3}}{{\sqrt[4]{8 b^{3}}}^{1 + 3}}$

Add $1$ and $3$ .

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

Rewrite ${\sqrt[4]{8 b^{3}}}^{4}$ as $8 b^{3}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{8 b^{3}}$ as ${(8 b^{3})}^{\frac{1}{4}}$ .

$\frac{\sqrt[4]{3 a^{2}} {\sqrt[4]{8 b^{3}}}^{3}}{{({(8 b^{3})}^{\frac{1}{4}})}^{4}}$

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

$\frac{\sqrt[4]{3 a^{2}} {\sqrt[4]{8 b^{3}}}^{3}}{{(8 b^{3})}^{\frac{1}{4} \cdot 4}}$

Combine $\frac{1}{4}$ and $4$ .

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

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{\sqrt[4]{3 a^{2}} {\sqrt[4]{8 b^{3}}}^{3}}{{(8 b^{3})}^{1}}$

Simplify.

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

Step 4

Simplify the numerator.

Rewrite ${\sqrt[4]{8 b^{3}}}^{3}$ as ${({(8 b^{3})}^{3})}^{\frac{1}{4}}$ .

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

Apply the product rule to $8 b^{3}$ .

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

Raise $8$ to the power of $3$ .

$\frac{\sqrt[4]{3 a^{2}} \sqrt[4]{512 {(b^{3})}^{3}}}{8 b^{3}}$

Multiply the exponents in ${(b^{3})}^{3}$ .

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

$\frac{\sqrt[4]{3 a^{2}} \sqrt[4]{512 b^{3 \cdot 3}}}{8 b^{3}}$

Multiply $3$ by $3$ .

$\frac{\sqrt[4]{3 a^{2}} \sqrt[4]{512 b^{9}}}{8 b^{3}}$

Rewrite $512 b^{9}$ as ${(4 b^{2})}^{4} \cdot (2 b)$ .

Factor $256$ out of $512$ .

$\frac{\sqrt[4]{3 a^{2}} \sqrt[4]{256 (2) b^{9}}}{8 b^{3}}$

Rewrite $256$ as $4^{4}$ .

$\frac{\sqrt[4]{3 a^{2}} \sqrt[4]{4^{4} \cdot 2 b^{9}}}{8 b^{3}}$

Factor out $b^{8}$ .

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

Rewrite $b^{8}$ as ${(b^{2})}^{4}$ .

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

Move $2$ .

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

Rewrite $4^{4} {(b^{2})}^{4}$ as ${(4 b^{2})}^{4}$ .

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

Add parentheses.

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

Pull terms out from under the radical.

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

Remove non-negative terms from the absolute value.

$\frac{\sqrt[4]{3 a^{2}} \cdot 4 b^{2} \sqrt[4]{2 b}}{8 b^{3}}$

Combine exponents.

Combine using the product rule for radicals.

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

Multiply $2$ by $3$ .

$\frac{4 b^{2} \sqrt[4]{6 a^{2} b}}{8 b^{3}}$

Step 5

Cancel the common factor of $4$ and $8$ .

Factor $4$ out of $4 b^{2} \sqrt[4]{6 a^{2} b}$ .

$\frac{4 (b^{2} \sqrt[4]{6 a^{2} b})}{8 b^{3}}$

Cancel the common factors.

Factor $4$ out of $8 b^{3}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 6

Cancel the common factor of $b^{2}$ and $b^{3}$ .

Factor $b^{2}$ out of $b^{2} \sqrt[4]{6 a^{2} b}$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the Radical Expression fourth root of (3a^2)/(8b^3)

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