Simplify the Radical Expression fourth root of 5^3* sixth root of 5

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

Step 1

Raise $5$ to the power of $3$ .

$\sqrt[4]{125} \cdot \sqrt[6]{5}$

Step 2

Multiply $\sqrt[4]{125} \sqrt[6]{5}$ .

Rewrite the expression using the least common index of $12$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{125}$ as $125^{\frac{1}{4}}$ .

$125^{\frac{1}{4}} \sqrt[6]{5}$

Rewrite $125^{\frac{1}{4}}$ as $125^{\frac{3}{12}}$ .

$125^{\frac{3}{12}} \sqrt[6]{5}$

Rewrite $125^{\frac{3}{12}}$ as $\sqrt[12]{125^{3}}$ .

$\sqrt[12]{125^{3}} \sqrt[6]{5}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{5}$ as $5^{\frac{1}{6}}$ .

$\sqrt[12]{125^{3}} \cdot 5^{\frac{1}{6}}$

Rewrite $5^{\frac{1}{6}}$ as $5^{\frac{2}{12}}$ .

$\sqrt[12]{125^{3}} \cdot 5^{\frac{2}{12}}$

Rewrite $5^{\frac{2}{12}}$ as $\sqrt[12]{5^{2}}$ .

$\sqrt[12]{125^{3}} \sqrt[12]{5^{2}}$

Combine using the product rule for radicals.

$\sqrt[12]{125^{3} \cdot 5^{2}}$

Rewrite $125$ as $5^{3}$ .

$\sqrt[12]{{(5^{3})}^{3} \cdot 5^{2}}$

Multiply the exponents in ${(5^{3})}^{3}$ .

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

$\sqrt[12]{5^{3 \cdot 3} \cdot 5^{2}}$

Multiply $3$ by $3$ .

$\sqrt[12]{5^{9} \cdot 5^{2}}$

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

$\sqrt[12]{5^{9 + 2}}$

Add $9$ and $2$ .

$\sqrt[12]{5^{11}}$

Step 3

Raise $5$ to the power of $11$ .

$\sqrt[12]{48828125}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\sqrt[12]{48828125}$

Decimal Form:

$4.37242636 \dots$