Solve (-ab^2c)^-1/a^-2bc^-1

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

Evaluate

$$-\frac{a}{b^{3}}$$

Solution Steps

Expand \left(\left(-a\right)b^{2}c\right)^{-1}.

$$\frac{\left(-a\right)^{-1}\left(b^{2}\right)^{-1}c^{-1}}{a^{-2}bc^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and -1 to get -2.

$$\frac{\left(-a\right)^{-1}b^{-2}c^{-1}}{a^{-2}bc^{-1}}$$

Cancel out \frac{1}{c} in both numerator and denominator.

$$\frac{b^{-2}\times \frac{1}{-a}}{a^{-2}b}$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

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

Express \frac{\frac{1}{-a}}{a^{-2}b^{3}} as a single fraction.

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

To multiply powers of the same base, add their exponents. Add 1 and -2 to get -1.

$$\frac{1}{-a^{-1}b^{3}}$$

Differentiate w.r.t. a

$$-\frac{1}{b^{3}}$$