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

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

Evaluate

$$\frac{a^{\frac{4}{3}}}{b^{3}}$$

Solution Steps

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

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

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

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

To raise a power to another power, multiply the exponents. Multiply 2 and -2 to get -4.

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

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

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

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

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

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

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

Calculate b to the power of 1 and get b.

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

To multiply powers of the same base, add their exponents. Add 2 and -\frac{2}{3} to get \frac{4}{3}.

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

To multiply powers of the same base, add their exponents. Add -4 and 1 to get -3.

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

Differentiate w.r.t. a

$$\frac{4\sqrt[3]{a}}{3b^{3}}$$