Simplify (-2b^-2c^3)^3

${(- 2 b^{- 2} c^{3})}^{3}$

Step 1

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

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

Step 2

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

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

Step 3

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

Move $2$ to the left of $c^{3}$ .

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

Step 6

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

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

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

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

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

Apply the product rule to $2 c^{3}$ .

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

Step 7

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

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

Step 8

Simplify the numerator.

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

$- \frac{8 {(c^{3})}^{3}}{{(b^{2})}^{3}}$

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

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

$- \frac{8 c^{3 \cdot 3}}{{(b^{2})}^{3}}$

Multiply $3$ by $3$ .

$- \frac{8 c^{9}}{{(b^{2})}^{3}}$

Step 9

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

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

$- \frac{8 c^{9}}{b^{2 \cdot 3}}$

Multiply $2$ by $3$ .

$- \frac{8 c^{9}}{b^{6}}$