Simplify ((a^2b^3)^4)/((ab^-2)^-3)

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

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

Multiply $2$ by $4$ .

$a^{8} {(b^{3})}^{4} {(a b^{- 2})}^{3}$

Step 4

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

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

$a^{8} b^{3 \cdot 4} {(a b^{- 2})}^{3}$

Multiply $3$ by $4$ .

$a^{8} b^{12} {(a b^{- 2})}^{3}$

Step 5

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

$a^{8} b^{12} {(a \frac{1}{b^{2}})}^{3}$

Step 6

Combine $a$ and $\frac{1}{b^{2}}$ .

$a^{8} b^{12} {(\frac{a}{b^{2}})}^{3}$

Step 7

Apply basic rules of exponents.

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

$a^{8} b^{12} \frac{a^{3}}{{(b^{2})}^{3}}$

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

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

$a^{8} b^{12} \frac{a^{3}}{b^{2 \cdot 3}}$

Multiply $2$ by $3$ .

$a^{8} b^{12} \frac{a^{3}}{b^{6}}$

Step 8

Cancel the common factor of $b^{6}$ .

Factor $b^{6}$ out of $a^{8} b^{12}$ .

$b^{6} (a^{8} b^{6}) \frac{a^{3}}{b^{6}}$

Cancel the common factor.

$b^{6} (a^{8} b^{6}) \frac{a^{3}}{b^{6}}$

Rewrite the expression.

$a^{8} b^{6} a^{3}$

Step 9

Multiply $a^{8}$ by $a^{3}$ by adding the exponents.

Move $a^{3}$ .

$a^{3} a^{8} b^{6}$

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

$a^{3 + 8} b^{6}$

Add $3$ and $8$ .

$a^{11} b^{6}$

Simplify ((a^2b^3)^4)/((ab^-2)^-3)

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