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

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

Evaluate

$$\frac{b|b|}{\sqrt{a}}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply -1 and -\frac{2}{3} to get \frac{2}{3}.

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

Multiply a^{-\frac{2}{3}} and a^{\frac{2}{3}} to get 1.

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

Expand \left(ab^{-2}\right)^{-\frac{1}{2}}.

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

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

$$ba^{-\frac{1}{2}}b^{1}$$

Calculate b to the power of 1 and get b.

$$ba^{-\frac{1}{2}}b$$

Multiply b and b to get b^{2}.

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

Differentiate w.r.t. b

$$\frac{2|b|}{\sqrt{a}}$$