Algebra Examples

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

Step 1

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

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Multiply $2$ by $2$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Multiply $3$ by $2$ .

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

Step 3

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

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

Step 4

Combine fractions.

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Step 4.1

Combine.

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

Step 4.2

Multiply $1$ by $1$ .

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

Step 5

Simplify the denominator.

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Step 5.1

Apply the product rule to $x^{5} y^{4}$ .

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

Step 5.2

Multiply the exponents in ${(x^{5})}^{3}$ .

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Step 5.2.1

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

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

Step 5.2.2

Multiply $5$ by $3$ .

$\frac{1}{x^{4} y^{6} x^{15} {(y^{4})}^{3}}$

Step 5.3

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

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Step 5.3.1

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

$\frac{1}{x^{4} y^{6} x^{15} y^{4 \cdot 3}}$

Step 5.3.2

Multiply $4$ by $3$ .

$\frac{1}{x^{4} y^{6} x^{15} y^{12}}$

Step 5.4

Combine exponents.

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Step 5.4.1

Multiply $x^{4}$ by $x^{15}$ by adding the exponents.

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Step 5.4.1.1

Move $x^{15}$ .

$\frac{1}{x^{15} x^{4} y^{6} y^{12}}$

Step 5.4.1.2

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

$\frac{1}{x^{15 + 4} y^{6} y^{12}}$

Step 5.4.1.3

Add $15$ and $4$ .

$\frac{1}{x^{19} y^{6} y^{12}}$

Step 5.4.2

Multiply $y^{6}$ by $y^{12}$ by adding the exponents.

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Step 5.4.2.1

Move $y^{12}$ .

$\frac{1}{x^{19} (y^{12} y^{6})}$

Step 5.4.2.2

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

$\frac{1}{x^{19} y^{12 + 6}}$

Step 5.4.2.3

Add $12$ and $6$ .

$\frac{1}{x^{19} y^{18}}$