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

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

Step 1

Move $b^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

Multiply $b$ by $b^{3}$ by adding the exponents.

Move $b^{3}$ .

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

Multiply $b^{3}$ by $b$ .

Raise $b$ to the power of $1$ .

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

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

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

Add $3$ and $1$ .

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

Step 3

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

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

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

Apply the product rule to $3 a$ .

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

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

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

Step 4

Raise $3$ to the power of $4$ .

$\frac{81 a^{4}}{2^{4} {(b^{4})}^{4}}$

Step 5

Simplify the denominator.

Raise $2$ to the power of $4$ .

$\frac{81 a^{4}}{16 {(b^{4})}^{4}}$

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

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

$\frac{81 a^{4}}{16 b^{4 \cdot 4}}$

Multiply $4$ by $4$ .

$\frac{81 a^{4}}{16 b^{16}}$