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

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

Step 1

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

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

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

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

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

Apply the product rule to $3 a^{3} b^{4}$ .

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

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

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

Step 2

Simplify the expression.

Raise $- 1$ to the power of $2$ .

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

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

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

Step 3

Simplify the numerator.

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

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

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

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

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

Multiply $3$ by $2$ .

$\frac{9 a^{6} {(b^{4})}^{2}}{2^{2}}$

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

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

$\frac{9 a^{6} b^{4 \cdot 2}}{2^{2}}$

Multiply $4$ by $2$ .

$\frac{9 a^{6} b^{8}}{2^{2}}$

Step 4

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

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