Precalculus Examples

${(- 125 x^{18} y^{24})}^{\frac{- 2}{3}}$

Step 1

Move the negative in front of the fraction.

${(- 125 x^{18} y^{24})}^{- \frac{2}{3}}$

Step 2

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

$\frac{1}{{(- 125 x^{18} y^{24})}^{\frac{2}{3}}}$

Step 3

Simplify the denominator.

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Step 3.1

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

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Step 3.1.1

Apply the product rule to $- 125 x^{18} y^{24}$ .

$\frac{1}{{(- 125 x^{18})}^{\frac{2}{3}} {(y^{24})}^{\frac{2}{3}}}$

Step 3.1.2

Apply the product rule to $- 125 x^{18}$ .

$\frac{1}{{(- 125)}^{\frac{2}{3}} {(x^{18})}^{\frac{2}{3}} {(y^{24})}^{\frac{2}{3}}}$

Step 3.2

Rewrite $- 125$ as ${(- 5)}^{3}$ .

$\frac{1}{{({(- 5)}^{3})}^{\frac{2}{3}} {(x^{18})}^{\frac{2}{3}} {(y^{24})}^{\frac{2}{3}}}$

Step 3.3

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

$\frac{1}{{(- 5)}^{3 (\frac{2}{3})} {(x^{18})}^{\frac{2}{3}} {(y^{24})}^{\frac{2}{3}}}$

Step 3.4

Cancel the common factor of $3$ .

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Step 3.4.1

Cancel the common factor.

$\frac{1}{{(- 5)}^{3 (\frac{2}{3})} {(x^{18})}^{\frac{2}{3}} {(y^{24})}^{\frac{2}{3}}}$

Step 3.4.2

Rewrite the expression.

$\frac{1}{{(- 5)}^{2} {(x^{18})}^{\frac{2}{3}} {(y^{24})}^{\frac{2}{3}}}$

Step 3.5

Raise $- 5$ to the power of $2$ .

$\frac{1}{25 {(x^{18})}^{\frac{2}{3}} {(y^{24})}^{\frac{2}{3}}}$

Step 3.6

Multiply the exponents in ${(x^{18})}^{\frac{2}{3}}$ .

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Step 3.6.1

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

$\frac{1}{25 x^{18 (\frac{2}{3})} {(y^{24})}^{\frac{2}{3}}}$

Step 3.6.2

Cancel the common factor of $3$ .

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Step 3.6.2.1

Factor $3$ out of $18$ .

$\frac{1}{25 x^{3 (6) \frac{2}{3}} {(y^{24})}^{\frac{2}{3}}}$

Step 3.6.2.2

Cancel the common factor.

$\frac{1}{25 x^{3 \cdot 6 \frac{2}{3}} {(y^{24})}^{\frac{2}{3}}}$

Step 3.6.2.3

Rewrite the expression.

$\frac{1}{25 x^{6 \cdot 2} {(y^{24})}^{\frac{2}{3}}}$

Step 3.6.3

Multiply $6$ by $2$ .

$\frac{1}{25 x^{12} {(y^{24})}^{\frac{2}{3}}}$

Step 3.7

Multiply the exponents in ${(y^{24})}^{\frac{2}{3}}$ .

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Step 3.7.1

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

$\frac{1}{25 x^{12} y^{24 (\frac{2}{3})}}$

Step 3.7.2

Cancel the common factor of $3$ .

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Step 3.7.2.1

Factor $3$ out of $24$ .

$\frac{1}{25 x^{12} y^{3 (8) \frac{2}{3}}}$

Step 3.7.2.2

Cancel the common factor.

$\frac{1}{25 x^{12} y^{3 \cdot 8 \frac{2}{3}}}$

Step 3.7.2.3

Rewrite the expression.

$\frac{1}{25 x^{12} y^{8 \cdot 2}}$

Step 3.7.3

Multiply $8$ by $2$ .

$\frac{1}{25 x^{12} y^{16}}$