Simplify ((2a^2b^3)/(x^3y^4))^-2

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

Step 1

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

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

Step 2

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

Apply the product rule to $\frac{x^{3} y^{4}}{2 a^{2} b^{3}}$ .

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

Apply the product rule to $x^{3} y^{4}$ .

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

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

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

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

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

Step 3

Simplify the numerator.

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

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

$\frac{x^{3 \cdot 2} {(y^{4})}^{2}}{2^{2} {(a^{2})}^{2} {(b^{3})}^{2}}$

Multiply $3$ by $2$ .

$\frac{x^{6} {(y^{4})}^{2}}{2^{2} {(a^{2})}^{2} {(b^{3})}^{2}}$

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

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

$\frac{x^{6} y^{4 \cdot 2}}{2^{2} {(a^{2})}^{2} {(b^{3})}^{2}}$

Multiply $4$ by $2$ .

$\frac{x^{6} y^{8}}{2^{2} {(a^{2})}^{2} {(b^{3})}^{2}}$

Step 4

Simplify the denominator.

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

$\frac{x^{6} y^{8}}{4 {(a^{2})}^{2} {(b^{3})}^{2}}$

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

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

$\frac{x^{6} y^{8}}{4 a^{2 \cdot 2} {(b^{3})}^{2}}$

Multiply $2$ by $2$ .

$\frac{x^{6} y^{8}}{4 a^{4} {(b^{3})}^{2}}$

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

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

$\frac{x^{6} y^{8}}{4 a^{4} b^{3 \cdot 2}}$

Multiply $3$ by $2$ .

$\frac{x^{6} y^{8}}{4 a^{4} b^{6}}$