Solve (frac{a^{-1}b^-2}{2ba^-1*4ab^2})^-1

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

Evaluate

$$8ab^{5}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

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

Multiply a^{-1} and a to get 1.

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

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

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

Multiply 2 and 4 to get 8.

$$\left(\frac{\frac{1}{a}}{8b^{5}}\right)^{-1}$$

Express \frac{\frac{1}{a}}{8b^{5}} as a single fraction.

$$\left(\frac{1}{a\times 8b^{5}}\right)^{-1}$$

To raise \frac{1}{a\times 8b^{5}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{1^{-1}}{\left(a\times 8b^{5}\right)^{-1}}$$

Calculate 1 to the power of -1 and get 1.

$$\frac{1}{\left(a\times 8b^{5}\right)^{-1}}$$

Expand \left(a\times 8b^{5}\right)^{-1}.

$$\frac{1}{a^{-1}\times 8^{-1}\left(b^{5}\right)^{-1}}$$

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

$$\frac{1}{a^{-1}\times 8^{-1}b^{-5}}$$

Calculate 8 to the power of -1 and get \frac{1}{8}.

$$\frac{1}{a^{-1}\times \frac{1}{8}b^{-5}}$$

Differentiate w.r.t. b

$$40ab^{4}$$