Solve (ab)^{2}c^{3}divfrac{a^frac{2{3}}b^frac{2{3}}}{3}c^-1div(-2c^2+bc)

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

Evaluate

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

Short Solution Steps

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

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

Divide a^{2}b^{2}c^{3} by \frac{a^{\frac{2}{3}}b^{\frac{2}{3}}}{3} by multiplying a^{2}b^{2}c^{3} by the reciprocal of \frac{a^{\frac{2}{3}}b^{\frac{2}{3}}}{3}.

$$\frac{\frac{a^{2}b^{2}c^{3}\times 3}{a^{\frac{2}{3}}b^{\frac{2}{3}}}c^{-1}}{-2c^{2}+bc}$$

Cancel out a^{\frac{2}{3}}b^{\frac{2}{3}} in both numerator and denominator.

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

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

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

Factor the expressions that are not already factored.

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

Cancel out c in both numerator and denominator.

$$\frac{3ca^{\frac{4}{3}}b^{\frac{4}{3}}}{b-2c}$$

Differentiate w.r.t. c

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