Algebra Examples

$(\sqrt[3]{32 m^{6}}) (\sqrt[3]{3125 m^{7}})$

Step 1

Rewrite $32 m^{6}$ as ${(2 m^{2})}^{3} \cdot 4$ .

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Step 1.1

Factor $8$ out of $32$ .

$\sqrt[3]{8 (4) m^{6}} \sqrt[3]{3125 m^{7}}$

Step 1.2

Rewrite $8$ as $2^{3}$ .

$\sqrt[3]{2^{3} \cdot 4 m^{6}} \sqrt[3]{3125 m^{7}}$

Step 1.3

Rewrite $m^{6}$ as ${(m^{2})}^{3}$ .

$\sqrt[3]{2^{3} \cdot 4 {(m^{2})}^{3}} \sqrt[3]{3125 m^{7}}$

Step 1.4

Move $4$ .

$\sqrt[3]{2^{3} {(m^{2})}^{3} \cdot 4} \sqrt[3]{3125 m^{7}}$

Step 1.5

Rewrite $2^{3} {(m^{2})}^{3}$ as ${(2 m^{2})}^{3}$ .

$\sqrt[3]{{(2 m^{2})}^{3} \cdot 4} \sqrt[3]{3125 m^{7}}$

Step 2

Pull terms out from under the radical.

$2 m^{2} \sqrt[3]{4} \sqrt[3]{3125 m^{7}}$

Step 3

Rewrite $3125 m^{7}$ as ${(5 m^{2})}^{3} \cdot (25 m)$ .

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Step 3.1

Factor $125$ out of $3125$ .

$2 m^{2} \sqrt[3]{4} \sqrt[3]{125 (25) m^{7}}$

Step 3.2

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

$2 m^{2} \sqrt[3]{4} \sqrt[3]{5^{3} \cdot 25 m^{7}}$

Step 3.3

Factor out $m^{6}$ .

$2 m^{2} \sqrt[3]{4} \sqrt[3]{5^{3} \cdot 25 (m^{6} m)}$

Step 3.4

Rewrite $m^{6}$ as ${(m^{2})}^{3}$ .

$2 m^{2} \sqrt[3]{4} \sqrt[3]{5^{3} \cdot 25 ({(m^{2})}^{3} m)}$

Step 3.5

Move $25$ .

$2 m^{2} \sqrt[3]{4} \sqrt[3]{5^{3} {(m^{2})}^{3} \cdot 25 m}$

Step 3.6

Rewrite $5^{3} {(m^{2})}^{3}$ as ${(5 m^{2})}^{3}$ .

$2 m^{2} \sqrt[3]{4} \sqrt[3]{{(5 m^{2})}^{3} \cdot 25 m}$

Step 3.7

Add parentheses.

$2 m^{2} \sqrt[3]{4} \sqrt[3]{{(5 m^{2})}^{3} \cdot (25 m)}$

Step 4

Pull terms out from under the radical.

$2 m^{2} \sqrt[3]{4} (5 m^{2} \sqrt[3]{25 m})$

Step 5

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

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Step 5.1

Move $m^{2}$ .

$2 (m^{2} m^{2}) \sqrt[3]{4} (5 \sqrt[3]{25 m})$

Step 5.2

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

$2 m^{2 + 2} \sqrt[3]{4} (5 \sqrt[3]{25 m})$

Step 5.3

Add $2$ and $2$ .

$2 m^{4} \sqrt[3]{4} (5 \sqrt[3]{25 m})$

Step 6

Multiply $2 m^{4} \sqrt[3]{4} (5 \sqrt[3]{25 m})$ .

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Step 6.1

Multiply $5$ by $2$ .

$10 m^{4} \sqrt[3]{4} \sqrt[3]{25 m}$

Step 6.2

Combine using the product rule for radicals.

$10 m^{4} \sqrt[3]{25 m \cdot 4}$

Step 6.3

Multiply $4$ by $25$ .

$10 m^{4} \sqrt[3]{100 m}$