Simplify square root of m^3 fifth root of m^4

$\sqrt{m^{3}} \sqrt[5]{m^{4}}$

Step 1

Factor out $m^{2}$ .

$\sqrt{m^{2} m} \sqrt[5]{m^{4}}$

Step 2

Pull terms out from under the radical.

$m \sqrt{m} \sqrt[5]{m^{4}}$

Step 3

Multiply $m \sqrt{m} \sqrt[5]{m^{4}}$ .

Rewrite the expression using the least common index of $10$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{m^{4}}$ as $m^{\frac{4}{5}}$ .

$m (m^{\frac{4}{5}} \sqrt{m})$

Rewrite $m^{\frac{4}{5}}$ as $m^{\frac{8}{10}}$ .

$m (m^{\frac{8}{10}} \sqrt{m})$

Rewrite $m^{\frac{8}{10}}$ as $\sqrt[10]{m^{8}}$ .

$m (\sqrt[10]{m^{8}} \sqrt{m})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{m}$ as $m^{\frac{1}{2}}$ .

$m (\sqrt[10]{m^{8}} m^{\frac{1}{2}})$

Rewrite $m^{\frac{1}{2}}$ as $m^{\frac{5}{10}}$ .

$m (\sqrt[10]{m^{8}} m^{\frac{5}{10}})$

Rewrite $m^{\frac{5}{10}}$ as $\sqrt[10]{m^{5}}$ .

$m (\sqrt[10]{m^{8}} \sqrt[10]{m^{5}})$

Combine using the product rule for radicals.

$m \sqrt[10]{m^{8} m^{5}}$

Multiply $m^{8}$ by $m^{5}$ by adding the exponents.

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

$m \sqrt[10]{m^{8 + 5}}$

Add $8$ and $5$ .

$m \sqrt[10]{m^{13}}$

Step 4

Factor out $m^{10}$ .

$m \sqrt[10]{m^{10} m^{3}}$

Step 5

Pull terms out from under the radical.

$m (m \sqrt[10]{m^{3}})$

Step 6

Multiply $m$ by $m$ by adding the exponents.

Move $m$ .

$m \cdot m \sqrt[10]{m^{3}}$

Multiply $m$ by $m$ .

$m^{2} \sqrt[10]{m^{3}}$