Simplify 10m^2 fourth root of 162m^7

$10 m^{2} \sqrt[4]{162 m^{7}}$

Step 1

Rewrite $162 m^{7}$ as ${(3 m)}^{4} \cdot (2 m^{3})$ .

Factor $81$ out of $162$ .

$10 m^{2} \sqrt[4]{81 (2) m^{7}}$

Rewrite $81$ as $3^{4}$ .

$10 m^{2} \sqrt[4]{3^{4} \cdot 2 m^{7}}$

Factor out $m^{4}$ .

$10 m^{2} \sqrt[4]{3^{4} \cdot 2 (m^{4} m^{3})}$

Move $2$ .

$10 m^{2} \sqrt[4]{3^{4} m^{4} \cdot 2 m^{3}}$

Rewrite $3^{4} m^{4}$ as ${(3 m)}^{4}$ .

$10 m^{2} \sqrt[4]{{(3 m)}^{4} \cdot 2 m^{3}}$

Add parentheses.

$10 m^{2} \sqrt[4]{{(3 m)}^{4} \cdot (2 m^{3})}$

Step 2

Pull terms out from under the radical.

$10 m^{2} (3 m \sqrt[4]{2 m^{3}})$

Step 3

Multiply $m^{2}$ by $m$ by adding the exponents.

Move $m$ .

$10 (m \cdot m^{2}) (3 \sqrt[4]{2 m^{3}})$

Multiply $m$ by $m^{2}$ .

Raise $m$ to the power of $1$ .

$10 (m^{1} m^{2}) (3 \sqrt[4]{2 m^{3}})$

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

$10 m^{1 + 2} (3 \sqrt[4]{2 m^{3}})$

Add $1$ and $2$ .

$10 m^{3} (3 \sqrt[4]{2 m^{3}})$

Step 4

Multiply $3$ by $10$ .

$30 m^{3} \sqrt[4]{2 m^{3}}$