Basic Math Examples

${(\sqrt[5]{32 m^{2} n^{10}})}^{3}$

Step 1

Rewrite $32 m^{2} n^{10}$ as ${(2 n^{2})}^{5} m^{2}$ .

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Step 1.1

Rewrite $32$ as $2^{5}$ .

${\sqrt[5]{2^{5} m^{2} n^{10}}}^{3}$

Step 1.2

Rewrite $n^{10}$ as ${(n^{2})}^{5}$ .

${\sqrt[5]{2^{5} m^{2} {(n^{2})}^{5}}}^{3}$

Step 1.3

Move $m^{2}$ .

${\sqrt[5]{2^{5} {(n^{2})}^{5} m^{2}}}^{3}$

Step 1.4

Rewrite $2^{5} {(n^{2})}^{5}$ as ${(2 n^{2})}^{5}$ .

${\sqrt[5]{{(2 n^{2})}^{5} m^{2}}}^{3}$

Step 2

Pull terms out from under the radical.

${(2 n^{2} \sqrt[5]{m^{2}})}^{3}$

Step 3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1

Apply the product rule to $2 n^{2} \sqrt[5]{m^{2}}$ .

${(2 n^{2})}^{3} {\sqrt[5]{m^{2}}}^{3}$

Step 3.2

Apply the product rule to $2 n^{2}$ .

$2^{3} {(n^{2})}^{3} {\sqrt[5]{m^{2}}}^{3}$

Step 4

Raise $2$ to the power of $3$ .

$8 {(n^{2})}^{3} {\sqrt[5]{m^{2}}}^{3}$

Step 5

Multiply the exponents in ${(n^{2})}^{3}$ .

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Step 5.1

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

$8 n^{2 \cdot 3} {\sqrt[5]{m^{2}}}^{3}$

Step 5.2

Multiply $2$ by $3$ .

$8 n^{6} {\sqrt[5]{m^{2}}}^{3}$

Step 6

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

$8 n^{6} \sqrt[5]{{(m^{2})}^{3}}$

Step 7

Multiply the exponents in ${(m^{2})}^{3}$ .

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Step 7.1

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

$8 n^{6} \sqrt[5]{m^{2 \cdot 3}}$

Step 7.2

Multiply $2$ by $3$ .

$8 n^{6} \sqrt[5]{m^{6}}$

Step 8

Factor out $m^{5}$ .

$8 n^{6} \sqrt[5]{m^{5} m}$

Step 9

Pull terms out from under the radical.

$8 n^{6} m \sqrt[5]{m}$