Calculus Examples

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

Step 1

Rewrite $- 4 m^{2}$ as ${({(- 1)}^{3})}^{3} \cdot (4 m^{2})$ .

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Step 1.1

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 1.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

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

Step 1.3

Rewrite $- 1$ as ${(- 1)}^{3}$ .

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

Step 1.4

Add parentheses.

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

Step 2

Pull terms out from under the radical.

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

Step 3

Simplify the expression.

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Step 3.1

Raise $- 1$ to the power of $3$ .

$- 1 \sqrt[3]{4 m^{2}} \cdot \sqrt[3]{2 m}$

Step 3.2

Rewrite $- 1 \sqrt[3]{4 m^{2}}$ as $- \sqrt[3]{4 m^{2}}$ .

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

Step 4

Multiply $- \sqrt[3]{4 m^{2}} \sqrt[3]{2 m}$ .

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Step 4.1

Combine using the product rule for radicals.

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

Step 4.2

Multiply $4$ by $2$ .

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

Step 4.3

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

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Step 4.3.1

Move $m^{2}$ .

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

Step 4.3.2

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

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Step 4.3.2.1

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

$- \sqrt[3]{8 (m^{2} m^{1})}$

Step 4.3.2.2

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

$- \sqrt[3]{8 m^{2 + 1}}$

Step 4.3.3

Add $2$ and $1$ .

$- \sqrt[3]{8 m^{3}}$

Step 5

Rewrite $8 m^{3}$ as ${(2 m)}^{3}$ .

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

Step 6

Pull terms out from under the radical, assuming real numbers.

$- (2 m)$

Step 7

Multiply $2$ by $- 1$ .

$- 2 m$