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

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

Evaluate

$$\frac{a^{7}}{b^{13}}$$

Solution Steps

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

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

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

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

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

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

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

$$\frac{\frac{\left(b^{5}\right)^{-2}}{\left(a^{\frac{5}{2}}\right)^{-2}}}{\left(\frac{\frac{1}{a}b^{\frac{9}{2}}}{a^{2}}\right)^{\frac{2}{3}}}$$

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

$$\frac{\frac{\left(b^{5}\right)^{-2}}{\left(a^{\frac{5}{2}}\right)^{-2}}}{\left(\frac{b^{\frac{9}{2}}}{a^{3}}\right)^{\frac{2}{3}}}$$

To raise \frac{b^{\frac{9}{2}}}{a^{3}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{\left(b^{5}\right)^{-2}}{\left(a^{\frac{5}{2}}\right)^{-2}}}{\frac{\left(b^{\frac{9}{2}}\right)^{\frac{2}{3}}}{\left(a^{3}\right)^{\frac{2}{3}}}}$$

Divide \frac{\left(b^{5}\right)^{-2}}{\left(a^{\frac{5}{2}}\right)^{-2}} by \frac{\left(b^{\frac{9}{2}}\right)^{\frac{2}{3}}}{\left(a^{3}\right)^{\frac{2}{3}}} by multiplying \frac{\left(b^{5}\right)^{-2}}{\left(a^{\frac{5}{2}}\right)^{-2}} by the reciprocal of \frac{\left(b^{\frac{9}{2}}\right)^{\frac{2}{3}}}{\left(a^{3}\right)^{\frac{2}{3}}}.

$$\frac{\left(b^{5}\right)^{-2}\left(a^{3}\right)^{\frac{2}{3}}}{\left(a^{\frac{5}{2}}\right)^{-2}\left(b^{\frac{9}{2}}\right)^{\frac{2}{3}}}$$

To raise a power to another power, multiply the exponents. Multiply 5 and -2 to get -10.

$$\frac{b^{-10}\left(a^{3}\right)^{\frac{2}{3}}}{\left(a^{\frac{5}{2}}\right)^{-2}\left(b^{\frac{9}{2}}\right)^{\frac{2}{3}}}$$

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

$$\frac{b^{-10}a^{2}}{\left(a^{\frac{5}{2}}\right)^{-2}\left(b^{\frac{9}{2}}\right)^{\frac{2}{3}}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{5}{2} and -2 to get -5.

$$\frac{b^{-10}a^{2}}{a^{-5}\left(b^{\frac{9}{2}}\right)^{\frac{2}{3}}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{9}{2} and \frac{2}{3} to get 3.

$$\frac{b^{-10}a^{2}}{a^{-5}b^{3}}$$

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

$$\frac{b^{-10}a^{7}}{b^{3}}$$

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

$$\frac{a^{7}}{b^{13}}$$

Differentiate w.r.t. b

$$-\frac{13a^{7}}{b^{14}}$$