Simplify fourth root of (5a^3)/(6b)

$\sqrt[4]{\frac{5 a^{3}}{6 b}}$

Step 1

Rewrite $\sqrt[4]{\frac{5 a^{3}}{6 b}}$ as $\frac{\sqrt[4]{5 a^{3}}}{\sqrt[4]{6 b}}$ .

$\frac{\sqrt[4]{5 a^{3}}}{\sqrt[4]{6 b}}$

Step 2

Multiply $\frac{\sqrt[4]{5 a^{3}}}{\sqrt[4]{6 b}}$ by $\frac{{\sqrt[4]{6 b}}^{3}}{{\sqrt[4]{6 b}}^{3}}$ .

$\frac{\sqrt[4]{5 a^{3}}}{\sqrt[4]{6 b}} \cdot \frac{{\sqrt[4]{6 b}}^{3}}{{\sqrt[4]{6 b}}^{3}}$

Step 3

Combine and simplify the denominator.

Multiply $\frac{\sqrt[4]{5 a^{3}}}{\sqrt[4]{6 b}}$ by $\frac{{\sqrt[4]{6 b}}^{3}}{{\sqrt[4]{6 b}}^{3}}$ .

$\frac{\sqrt[4]{5 a^{3}} {\sqrt[4]{6 b}}^{3}}{\sqrt[4]{6 b} {\sqrt[4]{6 b}}^{3}}$

Raise $\sqrt[4]{6 b}$ to the power of $1$ .

$\frac{\sqrt[4]{5 a^{3}} {\sqrt[4]{6 b}}^{3}}{{\sqrt[4]{6 b}}^{1} {\sqrt[4]{6 b}}^{3}}$

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

$\frac{\sqrt[4]{5 a^{3}} {\sqrt[4]{6 b}}^{3}}{{\sqrt[4]{6 b}}^{1 + 3}}$

Add $1$ and $3$ .

$\frac{\sqrt[4]{5 a^{3}} {\sqrt[4]{6 b}}^{3}}{{\sqrt[4]{6 b}}^{4}}$

Rewrite ${\sqrt[4]{6 b}}^{4}$ as $6 b$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{6 b}$ as ${(6 b)}^{\frac{1}{4}}$ .

$\frac{\sqrt[4]{5 a^{3}} {\sqrt[4]{6 b}}^{3}}{{({(6 b)}^{\frac{1}{4}})}^{4}}$

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

$\frac{\sqrt[4]{5 a^{3}} {\sqrt[4]{6 b}}^{3}}{{(6 b)}^{\frac{1}{4} \cdot 4}}$

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

$\frac{\sqrt[4]{5 a^{3}} {\sqrt[4]{6 b}}^{3}}{{(6 b)}^{\frac{4}{4}}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{\sqrt[4]{5 a^{3}} {\sqrt[4]{6 b}}^{3}}{{(6 b)}^{\frac{4}{4}}}$

Rewrite the expression.

$\frac{\sqrt[4]{5 a^{3}} {\sqrt[4]{6 b}}^{3}}{{(6 b)}^{1}}$

Simplify.

$\frac{\sqrt[4]{5 a^{3}} {\sqrt[4]{6 b}}^{3}}{6 b}$

Step 4

Simplify the numerator.

Rewrite ${\sqrt[4]{6 b}}^{3}$ as $\sqrt[4]{{(6 b)}^{3}}$ .

$\frac{\sqrt[4]{5 a^{3}} \sqrt[4]{{(6 b)}^{3}}}{6 b}$

Apply the product rule to $6 b$ .

$\frac{\sqrt[4]{5 a^{3}} \sqrt[4]{6^{3} b^{3}}}{6 b}$

Raise $6$ to the power of $3$ .

$\frac{\sqrt[4]{5 a^{3}} \sqrt[4]{216 b^{3}}}{6 b}$

Step 5

Simplify the numerator.

Combine using the product rule for radicals.

$\frac{\sqrt[4]{5 a^{3} (216 b^{3})}}{6 b}$

Multiply $216$ by $5$ .

$\frac{\sqrt[4]{1080 a^{3} b^{3}}}{6 b}$