Solve (a^3b)^-1*(ab^2)^frac{3{2}}

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

Evaluate

$$\frac{\left(|b|\right)^{3}}{ba^{\frac{3}{2}}}$$

Solution Steps

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

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

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

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

Expand \left(ab^{2}\right)^{\frac{3}{2}}.

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

To raise a power to another power, multiply the exponents. Multiply 2 and \frac{3}{2} to get 3.

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

To multiply powers of the same base, add their exponents. Add -3 and \frac{3}{2} to get -\frac{3}{2}.

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

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

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

Differentiate w.r.t. a

$$-\frac{3\left(|b|\right)^{3}}{2ba^{\frac{5}{2}}}$$