Simplify (2/3)^-1((3a)/(2b))^-2

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

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

Apply the product rule to $2 b$ .

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

Apply the product rule to $3 a$ .

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

Step 4

Combine.

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

Step 5

Cancel the common factors.

Factor $3$ out of $2 (3^{2} a^{2})$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 6

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

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

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

Cancel the common factors.

Cancel the common factor.

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

Rewrite the expression.

$\frac{2 b^{2}}{3 a^{2}}$