Simplify ((a^3b^2)/(7a^-4b^3))^-2

${(\frac{a^{3} b^{2}}{7 a^{- 4} b^{3}})}^{- 2}$

Step 1

Move $a^{- 4}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{a^{3} b^{2} a^{4}}{7 b^{3}})}^{- 2}$

Step 2

Multiply $a^{3}$ by $a^{4}$ by adding the exponents.

Move $a^{4}$ .

${(\frac{a^{4} a^{3} b^{2}}{7 b^{3}})}^{- 2}$

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

${(\frac{a^{4 + 3} b^{2}}{7 b^{3}})}^{- 2}$

Add $4$ and $3$ .

${(\frac{a^{7} b^{2}}{7 b^{3}})}^{- 2}$

Step 3

Cancel the common factor of $b^{2}$ and $b^{3}$ .

Factor $b^{2}$ out of $a^{7} b^{2}$ .

${(\frac{b^{2} a^{7}}{7 b^{3}})}^{- 2}$

Cancel the common factors.

Factor $b^{2}$ out of $7 b^{3}$ .

${(\frac{b^{2} a^{7}}{b^{2} (7 b)})}^{- 2}$

Cancel the common factor.

${(\frac{b^{2} a^{7}}{b^{2} (7 b)})}^{- 2}$

Rewrite the expression.

${(\frac{a^{7}}{7 b})}^{- 2}$

Step 4

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

${(\frac{7 b}{a^{7}})}^{2}$

Step 5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{7 b}{a^{7}}$ .

$\frac{{(7 b)}^{2}}{{(a^{7})}^{2}}$

Apply the product rule to $7 b$ .

$\frac{7^{2} b^{2}}{{(a^{7})}^{2}}$

Step 6

Raise $7$ to the power of $2$ .

$\frac{49 b^{2}}{{(a^{7})}^{2}}$

Step 7

Multiply the exponents in ${(a^{7})}^{2}$ .

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

$\frac{49 b^{2}}{a^{7 \cdot 2}}$

Multiply $7$ by $2$ .

$\frac{49 b^{2}}{a^{14}}$