Pre-Algebra Examples

${(\frac{x^{- 2} y^{3}}{- 2 x^{- 5} y^{7}})}^{- 3}$

Step 1

Move $x^{- 5}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{x^{- 2} y^{3} x^{5}}{- 2 y^{7}})}^{- 3}$

Step 2

Multiply $x^{- 2}$ by $x^{5}$ by adding the exponents.

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Step 2.1

Move $x^{5}$ .

${(\frac{x^{5} x^{- 2} y^{3}}{- 2 y^{7}})}^{- 3}$

Step 2.2

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

${(\frac{x^{5 - 2} y^{3}}{- 2 y^{7}})}^{- 3}$

Step 2.3

Subtract $2$ from $5$ .

${(\frac{x^{3} y^{3}}{- 2 y^{7}})}^{- 3}$

Step 3

Cancel the common factor of $y^{3}$ and $y^{7}$ .

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Step 3.1

Factor $y^{3}$ out of $x^{3} y^{3}$ .

${(\frac{y^{3} x^{3}}{- 2 y^{7}})}^{- 3}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $y^{3}$ out of $- 2 y^{7}$ .

${(\frac{y^{3} x^{3}}{y^{3} (- 2 y^{4})})}^{- 3}$

Step 3.2.2

Cancel the common factor.

${(\frac{y^{3} x^{3}}{y^{3} (- 2 y^{4})})}^{- 3}$

Step 3.2.3

Rewrite the expression.

${(\frac{x^{3}}{- 2 y^{4}})}^{- 3}$

Step 4

Move the negative in front of the fraction.

${(- \frac{x^{3}}{2 y^{4}})}^{- 3}$

Step 5

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

${(- \frac{2 y^{4}}{x^{3}})}^{3}$

Step 6

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

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Step 6.1

Apply the product rule to $- \frac{2 y^{4}}{x^{3}}$ .

${(- 1)}^{3} {(\frac{2 y^{4}}{x^{3}})}^{3}$

Step 6.2

Apply the product rule to $\frac{2 y^{4}}{x^{3}}$ .

${(- 1)}^{3} \frac{{(2 y^{4})}^{3}}{{(x^{3})}^{3}}$

Step 6.3

Apply the product rule to $2 y^{4}$ .

${(- 1)}^{3} \frac{2^{3} {(y^{4})}^{3}}{{(x^{3})}^{3}}$

Step 7

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

$- \frac{2^{3} {(y^{4})}^{3}}{{(x^{3})}^{3}}$

Step 8

Simplify the numerator.

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Step 8.1

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

$- \frac{8 {(y^{4})}^{3}}{{(x^{3})}^{3}}$

Step 8.2

Multiply the exponents in ${(y^{4})}^{3}$ .

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Step 8.2.1

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

$- \frac{8 y^{4 \cdot 3}}{{(x^{3})}^{3}}$

Step 8.2.2

Multiply $4$ by $3$ .

$- \frac{8 y^{12}}{{(x^{3})}^{3}}$

Step 9

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

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Step 9.1

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

$- \frac{8 y^{12}}{x^{3 \cdot 3}}$

Step 9.2

Multiply $3$ by $3$ .

$- \frac{8 y^{12}}{x^{9}}$