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

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

Step 1

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

Factor $b$ out of $- 3 a b^{2}$ .

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

Cancel the common factors.

Factor $b$ out of $4 b c^{4}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 2

Move the negative in front of the fraction.

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

Step 3

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

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

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

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

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

Apply the product rule to $3 a b$ .

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

Apply the product rule to $3 a$ .

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

Apply the product rule to $4 c^{4}$ .

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

Step 4

Evaluate the exponents.

Raise $- 1$ to the power of $3$ .

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

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

$- \frac{27 a^{3} b^{3}}{4^{3} {(c^{4})}^{3}}$

Step 5

Simplify the denominator.

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

$- \frac{27 a^{3} b^{3}}{64 {(c^{4})}^{3}}$

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

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

$- \frac{27 a^{3} b^{3}}{64 c^{4 \cdot 3}}$

Multiply $4$ by $3$ .

$- \frac{27 a^{3} b^{3}}{64 c^{12}}$