Calculus Examples

${(\frac{x^{3}}{- 27 y^{- 6}})}^{- \frac{2}{3}}$

Step 1

Move $y^{- 6}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{x^{3} y^{6}}{- 27})}^{- \frac{2}{3}}$

Step 2

Move the negative in front of the fraction.

${(- \frac{x^{3} y^{6}}{27})}^{- \frac{2}{3}}$

Step 3

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

${(- \frac{27}{x^{3} y^{6}})}^{\frac{2}{3}}$

Step 4

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

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Step 4.1

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

${(- 1)}^{\frac{2}{3}} {(\frac{27}{x^{3} y^{6}})}^{\frac{2}{3}}$

Step 4.2

Apply the product rule to $\frac{27}{x^{3} y^{6}}$ .

${(- 1)}^{\frac{2}{3}} \frac{27^{\frac{2}{3}}}{{(x^{3} y^{6})}^{\frac{2}{3}}}$

Step 4.3

Apply the product rule to $x^{3} y^{6}$ .

${(- 1)}^{\frac{2}{3}} \frac{27^{\frac{2}{3}}}{{(x^{3})}^{\frac{2}{3}} {(y^{6})}^{\frac{2}{3}}}$

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

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

${({(- 1)}^{3})}^{\frac{2}{3}} \frac{27^{\frac{2}{3}}}{{(x^{3})}^{\frac{2}{3}} {(y^{6})}^{\frac{2}{3}}}$

Step 5.2

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

${(- 1)}^{3 (\frac{2}{3})} \frac{27^{\frac{2}{3}}}{{(x^{3})}^{\frac{2}{3}} {(y^{6})}^{\frac{2}{3}}}$

Step 5.3

Cancel the common factor of $3$ .

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Step 5.3.1

Cancel the common factor.

${(- 1)}^{3 (\frac{2}{3})} \frac{27^{\frac{2}{3}}}{{(x^{3})}^{\frac{2}{3}} {(y^{6})}^{\frac{2}{3}}}$

Step 5.3.2

Rewrite the expression.

${(- 1)}^{2} \frac{27^{\frac{2}{3}}}{{(x^{3})}^{\frac{2}{3}} {(y^{6})}^{\frac{2}{3}}}$

Step 5.4

Simplify the expression.

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Step 5.4.1

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

$1 \frac{27^{\frac{2}{3}}}{{(x^{3})}^{\frac{2}{3}} {(y^{6})}^{\frac{2}{3}}}$

Step 5.4.2

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

$\frac{27^{\frac{2}{3}}}{{(x^{3})}^{\frac{2}{3}} {(y^{6})}^{\frac{2}{3}}}$

Step 6

Simplify the numerator.

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Step 6.1

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

$\frac{{(3^{3})}^{\frac{2}{3}}}{{(x^{3})}^{\frac{2}{3}} {(y^{6})}^{\frac{2}{3}}}$

Step 6.2

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

$\frac{3^{3 (\frac{2}{3})}}{{(x^{3})}^{\frac{2}{3}} {(y^{6})}^{\frac{2}{3}}}$

Step 6.3

Cancel the common factor of $3$ .

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Step 6.3.1

Cancel the common factor.

$\frac{3^{3 (\frac{2}{3})}}{{(x^{3})}^{\frac{2}{3}} {(y^{6})}^{\frac{2}{3}}}$

Step 6.3.2

Rewrite the expression.

$\frac{3^{2}}{{(x^{3})}^{\frac{2}{3}} {(y^{6})}^{\frac{2}{3}}}$

Step 6.4

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

$\frac{9}{{(x^{3})}^{\frac{2}{3}} {(y^{6})}^{\frac{2}{3}}}$

Step 7

Simplify the denominator.

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Step 7.1

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

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Step 7.1.1

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

$\frac{9}{x^{3 (\frac{2}{3})} {(y^{6})}^{\frac{2}{3}}}$

Step 7.1.2

Cancel the common factor of $3$ .

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Step 7.1.2.1

Cancel the common factor.

$\frac{9}{x^{3 (\frac{2}{3})} {(y^{6})}^{\frac{2}{3}}}$

Step 7.1.2.2

Rewrite the expression.

$\frac{9}{x^{2} {(y^{6})}^{\frac{2}{3}}}$

Step 7.2

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

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Step 7.2.1

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

$\frac{9}{x^{2} y^{6 (\frac{2}{3})}}$

Step 7.2.2

Cancel the common factor of $3$ .

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Step 7.2.2.1

Factor $3$ out of $6$ .

$\frac{9}{x^{2} y^{3 (2) \frac{2}{3}}}$

Step 7.2.2.2

Cancel the common factor.

$\frac{9}{x^{2} y^{3 \cdot 2 \frac{2}{3}}}$

Step 7.2.2.3

Rewrite the expression.

$\frac{9}{x^{2} y^{2 \cdot 2}}$

Step 7.2.3

Multiply $2$ by $2$ .

$\frac{9}{x^{2} y^{4}}$