Simplify ((2a^-2b^5)/(4a^2b^7))^-2

${(\frac{2 a^{- 2} b^{5}}{4 a^{2} b^{7}})}^{- 2}$

Step 1

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

${(\frac{2 b^{5}}{4 a^{2} b^{7} a^{2}})}^{- 2}$

Step 2

Multiply $a^{2}$ by $a^{2}$ by adding the exponents.

Move $a^{2}$ .

${(\frac{2 b^{5}}{4 (a^{2} a^{2}) b^{7}})}^{- 2}$

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

${(\frac{2 b^{5}}{4 a^{2 + 2} b^{7}})}^{- 2}$

Add $2$ and $2$ .

${(\frac{2 b^{5}}{4 a^{4} b^{7}})}^{- 2}$

Step 3

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2 b^{5}$ .

${(\frac{2 (b^{5})}{4 a^{4} b^{7}})}^{- 2}$

Cancel the common factors.

Factor $2$ out of $4 a^{4} b^{7}$ .

${(\frac{2 (b^{5})}{2 (2 a^{4} b^{7})})}^{- 2}$

Cancel the common factor.

${(\frac{2 b^{5}}{2 (2 a^{4} b^{7})})}^{- 2}$

Rewrite the expression.

${(\frac{b^{5}}{2 a^{4} b^{7}})}^{- 2}$

Step 4

Cancel the common factor of $b^{5}$ and $b^{7}$ .

Multiply by $1$ .

${(\frac{b^{5} \cdot 1}{2 a^{4} b^{7}})}^{- 2}$

Cancel the common factors.

Factor $b^{5}$ out of $2 a^{4} b^{7}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

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

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

Step 6

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

Apply the product rule to $2 a^{4} b^{2}$ .

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

Apply the product rule to $2 a^{4}$ .

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

Step 7

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

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

Step 8

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

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

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

Multiply $4$ by $2$ .

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

Step 9

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

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

$4 a^{8} b^{2 \cdot 2}$

Multiply $2$ by $2$ .

$4 a^{8} b^{4}$

Simplify ((2a^-2b^5)/(4a^2b^7))^-2

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