Algebra Examples

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

Step 1

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

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Multiply $3$ by $4$ .

$\frac{1}{x^{12} {(y^{2})}^{4}} \cdot {(x^{2} y^{- 3})}^{- 2}$

Step 2.3

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

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Step 2.3.1

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

$\frac{1}{x^{12} y^{2 \cdot 4}} \cdot {(x^{2} y^{- 3})}^{- 2}$

Step 2.3.2

Multiply $2$ by $4$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine fractions.

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Step 6.1

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

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

Step 6.2

Combine.

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

Step 6.3

Multiply ${(y^{3})}^{2}$ by $1$ .

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

Step 7

Simplify the denominator.

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Step 7.1

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

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Step 7.1.1

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

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

Step 7.1.2

Multiply $2$ by $2$ .

$\frac{{(y^{3})}^{2}}{x^{12} y^{8} x^{4}}$

Step 7.2

Multiply $x^{12}$ by $x^{4}$ by adding the exponents.

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Step 7.2.1

Move $x^{4}$ .

$\frac{{(y^{3})}^{2}}{x^{4} x^{12} y^{8}}$

Step 7.2.2

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

$\frac{{(y^{3})}^{2}}{x^{4 + 12} y^{8}}$

Step 7.2.3

Add $4$ and $12$ .

$\frac{{(y^{3})}^{2}}{x^{16} y^{8}}$

Step 8

Reduce the expression by cancelling the common factors.

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Step 8.1

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

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Step 8.1.1

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

$\frac{y^{3 \cdot 2}}{x^{16} y^{8}}$

Step 8.1.2

Multiply $3$ by $2$ .

$\frac{y^{6}}{x^{16} y^{8}}$

Step 8.2

Cancel the common factor of $y^{6}$ and $y^{8}$ .

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Step 8.2.1

Multiply by $1$ .

$\frac{y^{6} \cdot 1}{x^{16} y^{8}}$

Step 8.2.2

Cancel the common factors.

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Step 8.2.2.1

Factor $y^{6}$ out of $x^{16} y^{8}$ .

$\frac{y^{6} \cdot 1}{y^{6} (x^{16} y^{2})}$

Step 8.2.2.2

Cancel the common factor.

$\frac{y^{6} \cdot 1}{y^{6} (x^{16} y^{2})}$

Step 8.2.2.3

Rewrite the expression.

$\frac{1}{x^{16} y^{2}}$