Simplify -((b^2c)/(2d^3f^4))^-2

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

Step 1

Change the sign of the exponent by rewriting the base as its reciprocal.

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

Step 2

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

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

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

Apply the product rule to $2 d^{3} f^{4}$ .

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

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

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

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

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

Step 3

Simplify the numerator.

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

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

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

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

$- \frac{4 d^{3 \cdot 2} {(f^{4})}^{2}}{{(b^{2})}^{2} c^{2}}$

Multiply $3$ by $2$ .

$- \frac{4 d^{6} {(f^{4})}^{2}}{{(b^{2})}^{2} c^{2}}$

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

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

$- \frac{4 d^{6} f^{4 \cdot 2}}{{(b^{2})}^{2} c^{2}}$

Multiply $4$ by $2$ .

$- \frac{4 d^{6} f^{8}}{{(b^{2})}^{2} c^{2}}$

Step 4

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

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

$- \frac{4 d^{6} f^{8}}{b^{2 \cdot 2} c^{2}}$

Multiply $2$ by $2$ .

$- \frac{4 d^{6} f^{8}}{b^{4} c^{2}}$