Algebra Examples

$\frac{- 5 {(x^{- 2} y^{- 3})}^{3}}{7 {(x^{- 2} y^{3})}^{- 4}}$

Step 1

Move ${(x^{- 2} y^{3})}^{- 4}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{- 5 {(x^{- 2} y^{- 3})}^{3} {(x^{- 2} y^{3})}^{4}}{7}$

Step 2

Simplify the numerator.

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Step 2.1

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

$\frac{- 5 {(x^{- 2})}^{3} {(y^{- 3})}^{3} {(x^{- 2} y^{3})}^{4}}{7}$

Step 2.2

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

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Step 2.2.1

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

$\frac{- 5 x^{- 2 \cdot 3} {(y^{- 3})}^{3} {(x^{- 2} y^{3})}^{4}}{7}$

Step 2.2.2

Multiply $- 2$ by $3$ .

$\frac{- 5 x^{- 6} {(y^{- 3})}^{3} {(x^{- 2} y^{3})}^{4}}{7}$

Step 2.3

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

$\frac{- 5 \frac{1}{x^{6}} {(y^{- 3})}^{3} {(x^{- 2} y^{3})}^{4}}{7}$

Step 2.4

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

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Step 2.4.1

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

$\frac{- 5 \frac{1}{x^{6}} y^{- 3 \cdot 3} {(x^{- 2} y^{3})}^{4}}{7}$

Step 2.4.2

Multiply $- 3$ by $3$ .

$\frac{- 5 \frac{1}{x^{6}} y^{- 9} {(x^{- 2} y^{3})}^{4}}{7}$

Step 2.5

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

$\frac{- 5 \frac{1}{x^{6}} \frac{1}{y^{9}} {(x^{- 2} y^{3})}^{4}}{7}$

Step 2.6

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

$\frac{- 5 \frac{1}{x^{6}} \frac{1}{y^{9}} {(\frac{1}{x^{2}} y^{3})}^{4}}{7}$

Step 2.7

Combine $\frac{1}{x^{2}}$ and $y^{3}$ .

$\frac{- 5 \frac{1}{x^{6}} \frac{1}{y^{9}} {(\frac{y^{3}}{x^{2}})}^{4}}{7}$

Step 2.8

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

$\frac{- 5 \frac{1}{x^{6}} \frac{1}{y^{9}} \frac{{(y^{3})}^{4}}{{(x^{2})}^{4}}}{7}$

Step 2.9

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

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Step 2.9.1

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

$\frac{- 5 \frac{1}{x^{6}} \frac{1}{y^{9}} \frac{y^{3 \cdot 4}}{{(x^{2})}^{4}}}{7}$

Step 2.9.2

Multiply $3$ by $4$ .

$\frac{- 5 \frac{1}{x^{6}} \frac{1}{y^{9}} \frac{y^{12}}{{(x^{2})}^{4}}}{7}$

Step 2.10

Multiply the exponents in ${(x^{2})}^{4}$ .

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Step 2.10.1

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

$\frac{- 5 \frac{1}{x^{6}} \frac{1}{y^{9}} \frac{y^{12}}{x^{2 \cdot 4}}}{7}$

Step 2.10.2

Multiply $2$ by $4$ .

$\frac{- 5 \frac{1}{x^{6}} \frac{1}{y^{9}} \frac{y^{12}}{x^{8}}}{7}$

Step 2.11

Combine exponents.

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Step 2.11.1

Combine $- 5$ and $\frac{1}{x^{6}}$ .

$\frac{\frac{- 5}{x^{6}} \cdot \frac{1}{y^{9}} \frac{y^{12}}{x^{8}}}{7}$

Step 2.11.2

Multiply $\frac{- 5}{x^{6}}$ by $\frac{1}{y^{9}}$ .

$\frac{\frac{- 5}{x^{6} y^{9}} \cdot \frac{y^{12}}{x^{8}}}{7}$

Step 2.11.3

Multiply $\frac{- 5}{x^{6} y^{9}}$ by $\frac{y^{12}}{x^{8}}$ .

$\frac{\frac{- 5 y^{12}}{x^{6} y^{9} x^{8}}}{7}$

Step 2.11.4

Multiply $x^{6}$ by $x^{8}$ by adding the exponents.

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Step 2.11.4.1

Move $x^{8}$ .

$\frac{\frac{- 5 y^{12}}{x^{8} x^{6} y^{9}}}{7}$

Step 2.11.4.2

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

$\frac{\frac{- 5 y^{12}}{x^{8 + 6} y^{9}}}{7}$

Step 2.11.4.3

Add $8$ and $6$ .

$\frac{\frac{- 5 y^{12}}{x^{14} y^{9}}}{7}$

Step 2.12

Reduce the expression $\frac{- 5 y^{12}}{x^{14} y^{9}}$ by cancelling the common factors.

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Step 2.12.1

Factor $y^{9}$ out of $- 5 y^{12}$ .

$\frac{\frac{y^{9} (- 5 y^{3})}{x^{14} y^{9}}}{7}$

Step 2.12.2

Factor $y^{9}$ out of $x^{14} y^{9}$ .

$\frac{\frac{y^{9} (- 5 y^{3})}{y^{9} x^{14}}}{7}$

Step 2.12.3

Cancel the common factor.

$\frac{\frac{y^{9} (- 5 y^{3})}{y^{9} x^{14}}}{7}$

Step 2.12.4

Rewrite the expression.

$\frac{\frac{- 5 y^{3}}{x^{14}}}{7}$

Step 2.13

Move the negative in front of the fraction.

$\frac{- \frac{5 y^{3}}{x^{14}}}{7}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$- \frac{5 y^{3}}{x^{14}} \cdot \frac{1}{7}$

Step 4

Multiply $\frac{1}{7}$ by $\frac{5 y^{3}}{x^{14}}$ .

$- \frac{5 y^{3}}{7 x^{14}}$