Solve {left(3a^-2b/2a^3b^-2right)}^-1

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

Evaluate

$$\frac{2a^{5}}{3b^{3}}$$

Solution Steps

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

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

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

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

To raise \frac{3b^{3}}{2a^{5}} to a power, raise both numerator and denominator to the power and then divide.

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

Expand \left(3b^{3}\right)^{-1}.

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

To raise a power to another power, multiply the exponents. Multiply 3 and -1 to get -3.

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

Calculate 3 to the power of -1 and get \frac{1}{3}.

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

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

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

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

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

Calculate 2 to the power of -1 and get \frac{1}{2}.

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

Differentiate w.r.t. b

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