Simplify (a^2b^3)^-2(a^5b^4)^-3

${(a^{2} b^{3})}^{- 2} {(a^{5} b^{4})}^{- 3}$

Step 1

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

$\frac{1}{{(a^{2} b^{3})}^{2}} {(a^{5} b^{4})}^{- 3}$

Step 2

Simplify the denominator.

Apply the product rule to $a^{2} b^{3}$ .

$\frac{1}{{(a^{2})}^{2} {(b^{3})}^{2}} {(a^{5} b^{4})}^{- 3}$

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

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

$\frac{1}{a^{2 \cdot 2} {(b^{3})}^{2}} {(a^{5} b^{4})}^{- 3}$

Multiply $2$ by $2$ .

$\frac{1}{a^{4} {(b^{3})}^{2}} {(a^{5} b^{4})}^{- 3}$

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

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

$\frac{1}{a^{4} b^{3 \cdot 2}} {(a^{5} b^{4})}^{- 3}$

Multiply $3$ by $2$ .

$\frac{1}{a^{4} b^{6}} {(a^{5} b^{4})}^{- 3}$

Step 3

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

$\frac{1}{a^{4} b^{6}} \cdot \frac{1}{{(a^{5} b^{4})}^{3}}$

Step 4

Combine fractions.

Combine.

$\frac{1 \cdot 1}{a^{4} b^{6} {(a^{5} b^{4})}^{3}}$

Multiply $1$ by $1$ .

$\frac{1}{a^{4} b^{6} {(a^{5} b^{4})}^{3}}$

Step 5

Simplify the denominator.

Apply the product rule to $a^{5} b^{4}$ .

$\frac{1}{a^{4} b^{6} {(a^{5})}^{3} {(b^{4})}^{3}}$

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

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

$\frac{1}{a^{4} b^{6} a^{5 \cdot 3} {(b^{4})}^{3}}$

Multiply $5$ by $3$ .

$\frac{1}{a^{4} b^{6} a^{15} {(b^{4})}^{3}}$

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

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

$\frac{1}{a^{4} b^{6} a^{15} b^{4 \cdot 3}}$

Multiply $4$ by $3$ .

$\frac{1}{a^{4} b^{6} a^{15} b^{12}}$

Combine exponents.

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

Move $a^{15}$ .

$\frac{1}{a^{15} a^{4} b^{6} b^{12}}$

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

$\frac{1}{a^{15 + 4} b^{6} b^{12}}$

Add $15$ and $4$ .

$\frac{1}{a^{19} b^{6} b^{12}}$

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

Move $b^{12}$ .

$\frac{1}{a^{19} (b^{12} b^{6})}$

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

$\frac{1}{a^{19} b^{12 + 6}}$

Add $12$ and $6$ .

$\frac{1}{a^{19} b^{18}}$

Simplify (a^2b^3)^-2(a^5b^4)^-3

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