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

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

Step 1

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

${(3 \frac{1}{b^{2}})}^{2} {(a^{2} b^{4})}^{3}$

Step 2

Combine $3$ and $\frac{1}{b^{2}}$ .

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

Step 3

Simplify the expression.

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

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

Raise $3$ to the power of $2$ .

$\frac{9}{{(b^{2})}^{2}} {(a^{2} b^{4})}^{3}$

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

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

$\frac{9}{b^{2 \cdot 2}} {(a^{2} b^{4})}^{3}$

Multiply $2$ by $2$ .

$\frac{9}{b^{4}} {(a^{2} b^{4})}^{3}$

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

$\frac{9}{b^{4}} ({(a^{2})}^{3} {(b^{4})}^{3})$

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

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

$\frac{9}{b^{4}} (a^{2 \cdot 3} {(b^{4})}^{3})$

Multiply $2$ by $3$ .

$\frac{9}{b^{4}} (a^{6} {(b^{4})}^{3})$

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

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

$\frac{9}{b^{4}} (a^{6} b^{4 \cdot 3})$

Multiply $4$ by $3$ .

$\frac{9}{b^{4}} (a^{6} b^{12})$

Step 4

Cancel the common factor of $b^{4}$ .

Factor $b^{4}$ out of $a^{6} b^{12}$ .

$\frac{9}{b^{4}} (b^{4} (a^{6} b^{8}))$

Cancel the common factor.

$\frac{9}{b^{4}} (b^{4} (a^{6} b^{8}))$

Rewrite the expression.

$9 (a^{6} b^{8})$

$9 a^{6} b^{8}$