Solve (frac{(-a^{4}b)^frac{2{3}}(ab)^frac{1{3}}}{-a^8b^8})

$$( \frac { ( - a ^ { 4 } b ) ^ { \frac { 2 } { 3 } } ( a b ) ^ { \frac { 1 } { 3 } } } { - a ^ { 8 } b ^ { 8 } } )$$

Evaluate

$$-\frac{1}{a^{5}b^{7}}$$

Short Solution Steps

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

$$\frac{\left(-a^{4}\right)^{\frac{2}{3}}b^{\frac{2}{3}}\left(ab\right)^{\frac{1}{3}}}{\left(-a^{8}\right)b^{8}}$$

Expand \left(ab\right)^{\frac{1}{3}}.

$$\frac{\left(-a^{4}\right)^{\frac{2}{3}}b^{\frac{2}{3}}a^{\frac{1}{3}}b^{\frac{1}{3}}}{\left(-a^{8}\right)b^{8}}$$

To multiply powers of the same base, add their exponents. Add \frac{2}{3} and \frac{1}{3} to get 1.

$$\frac{\left(-a^{4}\right)^{\frac{2}{3}}b^{1}a^{\frac{1}{3}}}{\left(-a^{8}\right)b^{8}}$$

Cancel out b in both numerator and denominator.

$$\frac{\sqrt[3]{a}\left(-a^{4}\right)^{\frac{2}{3}}}{b^{7}\left(-a^{8}\right)}$$

Factor the expressions that are not already factored.

$$\frac{\sqrt[3]{a}\left(-a^{4}\right)^{\frac{2}{3}}}{-b^{7}a^{8}}$$

Cancel out \sqrt[3]{a} in both numerator and denominator.

$$\frac{\left(-a^{4}\right)^{\frac{2}{3}}}{-b^{7}a^{\frac{23}{3}}}$$

Expand the expression.

$$\frac{a^{\frac{8}{3}}}{-b^{7}a^{\frac{23}{3}}}$$

Cancel out a^{\frac{8}{3}} in both numerator and denominator.

$$\frac{1}{-a^{5}b^{7}}$$

Differentiate w.r.t. a

$$\frac{5}{a^{6}b^{7}}$$