Simplify ( cube root of 3a)/( cube root of 5c)

$\frac{\sqrt[3]{3 a}}{\sqrt[3]{5 c}}$

Step 1

Multiply $\frac{\sqrt[3]{3 a}}{\sqrt[3]{5 c}}$ by $\frac{{\sqrt[3]{5 c}}^{2}}{{\sqrt[3]{5 c}}^{2}}$ .

$\frac{\sqrt[3]{3 a}}{\sqrt[3]{5 c}} \cdot \frac{{\sqrt[3]{5 c}}^{2}}{{\sqrt[3]{5 c}}^{2}}$

Step 2

Combine and simplify the denominator.

Multiply $\frac{\sqrt[3]{3 a}}{\sqrt[3]{5 c}}$ by $\frac{{\sqrt[3]{5 c}}^{2}}{{\sqrt[3]{5 c}}^{2}}$ .

$\frac{\sqrt[3]{3 a} {\sqrt[3]{5 c}}^{2}}{\sqrt[3]{5 c} {\sqrt[3]{5 c}}^{2}}$

Raise $\sqrt[3]{5 c}$ to the power of $1$ .

$\frac{\sqrt[3]{3 a} {\sqrt[3]{5 c}}^{2}}{{\sqrt[3]{5 c}}^{1} {\sqrt[3]{5 c}}^{2}}$

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

$\frac{\sqrt[3]{3 a} {\sqrt[3]{5 c}}^{2}}{{\sqrt[3]{5 c}}^{1 + 2}}$

Add $1$ and $2$ .

$\frac{\sqrt[3]{3 a} {\sqrt[3]{5 c}}^{2}}{{\sqrt[3]{5 c}}^{3}}$

Rewrite ${\sqrt[3]{5 c}}^{3}$ as $5 c$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5 c}$ as ${(5 c)}^{\frac{1}{3}}$ .

$\frac{\sqrt[3]{3 a} {\sqrt[3]{5 c}}^{2}}{{({(5 c)}^{\frac{1}{3}})}^{3}}$

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

$\frac{\sqrt[3]{3 a} {\sqrt[3]{5 c}}^{2}}{{(5 c)}^{\frac{1}{3} \cdot 3}}$

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

$\frac{\sqrt[3]{3 a} {\sqrt[3]{5 c}}^{2}}{{(5 c)}^{\frac{3}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{\sqrt[3]{3 a} {\sqrt[3]{5 c}}^{2}}{{(5 c)}^{\frac{3}{3}}}$

Rewrite the expression.

$\frac{\sqrt[3]{3 a} {\sqrt[3]{5 c}}^{2}}{{(5 c)}^{1}}$

Simplify.

$\frac{\sqrt[3]{3 a} {\sqrt[3]{5 c}}^{2}}{5 c}$

Step 3

Simplify the numerator.

Rewrite ${\sqrt[3]{5 c}}^{2}$ as $\sqrt[3]{{(5 c)}^{2}}$ .

$\frac{\sqrt[3]{3 a} \sqrt[3]{{(5 c)}^{2}}}{5 c}$

Apply the product rule to $5 c$ .

$\frac{\sqrt[3]{3 a} \sqrt[3]{5^{2} c^{2}}}{5 c}$

Raise $5$ to the power of $2$ .

$\frac{\sqrt[3]{3 a} \sqrt[3]{25 c^{2}}}{5 c}$

Step 4

Simplify the numerator.

Combine using the product rule for radicals.

$\frac{\sqrt[3]{3 a (25 c^{2})}}{5 c}$

Multiply $25$ by $3$ .

$\frac{\sqrt[3]{75 a c^{2}}}{5 c}$