Trigonometry Examples

${(- 27 m^{- 3} n^{6})}^{- \frac{2}{3}}$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(- 27 \frac{1}{m^{3}} n^{6})}^{- \frac{2}{3}}$

Step 2

Combine $- 27$ and $\frac{1}{m^{3}}$ .

${(\frac{- 27}{m^{3}} n^{6})}^{- \frac{2}{3}}$

Step 3

Move the negative in front of the fraction.

${(- \frac{27}{m^{3}} n^{6})}^{- \frac{2}{3}}$

Step 4

Combine $n^{6}$ and $\frac{27}{m^{3}}$ .

${(- \frac{n^{6} \cdot 27}{m^{3}})}^{- \frac{2}{3}}$

Step 5

Move $27$ to the left of $n^{6}$ .

${(- \frac{27 \cdot n^{6}}{m^{3}})}^{- \frac{2}{3}}$

Step 6

Change the sign of the exponent by rewriting the base as its reciprocal.

${(- \frac{m^{3}}{27 n^{6}})}^{\frac{2}{3}}$

Step 7

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

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Step 7.1

Apply the product rule to $- \frac{m^{3}}{27 n^{6}}$ .

${(- 1)}^{\frac{2}{3}} {(\frac{m^{3}}{27 n^{6}})}^{\frac{2}{3}}$

Step 7.2

Apply the product rule to $\frac{m^{3}}{27 n^{6}}$ .

${(- 1)}^{\frac{2}{3}} \frac{{(m^{3})}^{\frac{2}{3}}}{{(27 n^{6})}^{\frac{2}{3}}}$

Step 7.3

Apply the product rule to $27 n^{6}$ .

${(- 1)}^{\frac{2}{3}} \frac{{(m^{3})}^{\frac{2}{3}}}{27^{\frac{2}{3}} {(n^{6})}^{\frac{2}{3}}}$

Step 8

Reduce the expression by cancelling the common factors.

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Step 8.1

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

${({(- 1)}^{3})}^{\frac{2}{3}} \frac{{(m^{3})}^{\frac{2}{3}}}{27^{\frac{2}{3}} {(n^{6})}^{\frac{2}{3}}}$

Step 8.2

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

${(- 1)}^{3 (\frac{2}{3})} \frac{{(m^{3})}^{\frac{2}{3}}}{27^{\frac{2}{3}} {(n^{6})}^{\frac{2}{3}}}$

Step 8.3

Cancel the common factor of $3$ .

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Step 8.3.1

Cancel the common factor.

${(- 1)}^{3 (\frac{2}{3})} \frac{{(m^{3})}^{\frac{2}{3}}}{27^{\frac{2}{3}} {(n^{6})}^{\frac{2}{3}}}$

Step 8.3.2

Rewrite the expression.

${(- 1)}^{2} \frac{{(m^{3})}^{\frac{2}{3}}}{27^{\frac{2}{3}} {(n^{6})}^{\frac{2}{3}}}$

Step 8.4

Simplify the expression.

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Step 8.4.1

Raise $- 1$ to the power of $2$ .

$1 \frac{{(m^{3})}^{\frac{2}{3}}}{27^{\frac{2}{3}} {(n^{6})}^{\frac{2}{3}}}$

Step 8.4.2

Multiply $\frac{{(m^{3})}^{\frac{2}{3}}}{27^{\frac{2}{3}} {(n^{6})}^{\frac{2}{3}}}$ by $1$ .

$\frac{{(m^{3})}^{\frac{2}{3}}}{27^{\frac{2}{3}} {(n^{6})}^{\frac{2}{3}}}$

Step 8.4.3

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

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Step 8.4.3.1

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

$\frac{m^{3 (\frac{2}{3})}}{27^{\frac{2}{3}} {(n^{6})}^{\frac{2}{3}}}$

Step 8.4.3.2

Cancel the common factor of $3$ .

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Step 8.4.3.2.1

Cancel the common factor.

$\frac{m^{3 (\frac{2}{3})}}{27^{\frac{2}{3}} {(n^{6})}^{\frac{2}{3}}}$

Step 8.4.3.2.2

Rewrite the expression.

$\frac{m^{2}}{27^{\frac{2}{3}} {(n^{6})}^{\frac{2}{3}}}$

Step 9

Simplify the denominator.

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Step 9.1

Rewrite $27$ as $3^{3}$ .

$\frac{m^{2}}{{(3^{3})}^{\frac{2}{3}} {(n^{6})}^{\frac{2}{3}}}$

Step 9.2

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

$\frac{m^{2}}{3^{3 (\frac{2}{3})} {(n^{6})}^{\frac{2}{3}}}$

Step 9.3

Cancel the common factor of $3$ .

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Step 9.3.1

Cancel the common factor.

$\frac{m^{2}}{3^{3 (\frac{2}{3})} {(n^{6})}^{\frac{2}{3}}}$

Step 9.3.2

Rewrite the expression.

$\frac{m^{2}}{3^{2} {(n^{6})}^{\frac{2}{3}}}$

Step 9.4

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

$\frac{m^{2}}{9 {(n^{6})}^{\frac{2}{3}}}$

Step 9.5

Multiply the exponents in ${(n^{6})}^{\frac{2}{3}}$ .

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Step 9.5.1

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

$\frac{m^{2}}{9 n^{6 (\frac{2}{3})}}$

Step 9.5.2

Cancel the common factor of $3$ .

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Step 9.5.2.1

Factor $3$ out of $6$ .

$\frac{m^{2}}{9 n^{3 (2) \frac{2}{3}}}$

Step 9.5.2.2

Cancel the common factor.

$\frac{m^{2}}{9 n^{3 \cdot 2 \frac{2}{3}}}$

Step 9.5.2.3

Rewrite the expression.

$\frac{m^{2}}{9 n^{2 \cdot 2}}$

Step 9.5.3

Multiply $2$ by $2$ .

$\frac{m^{2}}{9 n^{4}}$