Trigonometry Examples

$\frac{\sqrt[3]{- 162 m^{6}}}{\sqrt[3]{3 m^{- 3}}}$

Step 1

Combine $\sqrt[3]{- 162 m^{6}}$ and $\sqrt[3]{3 m^{- 3}}$ into a single radical.

$\sqrt[3]{\frac{- 162 m^{6}}{3 m^{- 3}}}$

Step 2

Reduce the expression $\frac{- 162 m^{6}}{3 m^{- 3}}$ by cancelling the common factors.

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Step 2.1

Factor $3$ out of $- 162 m^{6}$ .

$\sqrt[3]{\frac{3 (- 54 m^{6})}{3 m^{- 3}}}$

Step 2.2

Factor $3$ out of $3 m^{- 3}$ .

$\sqrt[3]{\frac{3 (- 54 m^{6})}{3 (m^{- 3})}}$

Step 2.3

Cancel the common factor.

$\sqrt[3]{\frac{3 (- 54 m^{6})}{3 m^{- 3}}}$

Step 2.4

Rewrite the expression.

$\sqrt[3]{\frac{- 54 m^{6}}{m^{- 3}}}$

Step 3

Move $m^{- 3}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\sqrt[3]{- 54 m^{6} m^{3}}$

Step 4

Multiply $m^{6}$ by $m^{3}$ by adding the exponents.

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Step 4.1

Move $m^{3}$ .

$\sqrt[3]{- 54 (m^{3} m^{6})}$

Step 4.2

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

$\sqrt[3]{- 54 m^{3 + 6}}$

Step 4.3

Add $3$ and $6$ .

$\sqrt[3]{- 54 m^{9}}$

Step 5

Rewrite $- 54 m^{9}$ as ${(- 3 m^{3})}^{3} \cdot 2$ .

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Step 5.1

Factor $- 27$ out of $- 54$ .

$\sqrt[3]{- 27 (2) m^{9}}$

Step 5.2

Rewrite $- 27$ as ${(- 3)}^{3}$ .

$\sqrt[3]{{(- 3)}^{3} \cdot 2 m^{9}}$

Step 5.3

Rewrite $m^{9}$ as ${(m^{3})}^{3}$ .

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

Step 5.4

Move $2$ .

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

Step 5.5

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

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

Step 6

Pull terms out from under the radical.

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