Solve frac{2a^{-3}b^4}{(4a^2b^-5)^-1}

$$\frac { 2 a ^ { - 3 } b ^ { 4 } } { ( 4 a ^ { 2 } b ^ { - 5 } ) ^ { - 1 } }$$

Evaluate

$$\frac{8}{ab}$$

Solution Steps

Expand \left(4a^{2}b^{-5}\right)^{-1}.

$$\frac{2a^{-3}b^{4}}{4^{-1}\left(a^{2}\right)^{-1}\left(b^{-5}\right)^{-1}}$$

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

$$\frac{2a^{-3}b^{4}}{4^{-1}a^{-2}\left(b^{-5}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply -5 and -1 to get 5.

$$\frac{2a^{-3}b^{4}}{4^{-1}a^{-2}b^{5}}$$

Calculate 4 to the power of -1 and get \frac{1}{4}.

$$\frac{2a^{-3}b^{4}}{\frac{1}{4}a^{-2}b^{5}}$$

Cancel out b^{4} in both numerator and denominator.

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

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

$$\frac{2}{\frac{1}{4}ba^{1}}$$

Calculate a to the power of 1 and get a.

$$\frac{2}{\frac{1}{4}ba}$$

Differentiate w.r.t. b

$$-\frac{8}{ab^{2}}$$