Solve frac{(32a^{15}c^{-5})^1/3}{(-27a^6c^-3)^1/3}

$$\frac { ( 32 a ^ { 15 } c ^ { - 5 } ) ^ { 1 / 3 } } { ( - 27 a ^ { 6 } c ^ { - 3 } ) ^ { 1 / 3 } }$$

Evaluate

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

Short Solution Steps

Expand \left(32a^{15}c^{-5}\right)^{\frac{1}{3}}.

$$\frac{32^{\frac{1}{3}}\left(a^{15}\right)^{\frac{1}{3}}\left(c^{-5}\right)^{\frac{1}{3}}}{\left(-27a^{6}c^{-3}\right)^{\frac{1}{3}}}$$

To raise a power to another power, multiply the exponents. Multiply 15 and \frac{1}{3} to get 5.

$$\frac{32^{\frac{1}{3}}a^{5}\left(c^{-5}\right)^{\frac{1}{3}}}{\left(-27a^{6}c^{-3}\right)^{\frac{1}{3}}}$$

To raise a power to another power, multiply the exponents. Multiply -5 and \frac{1}{3} to get -\frac{5}{3}.

$$\frac{32^{\frac{1}{3}}a^{5}c^{-\frac{5}{3}}}{\left(-27a^{6}c^{-3}\right)^{\frac{1}{3}}}$$

Expand \left(-27a^{6}c^{-3}\right)^{\frac{1}{3}}.

$$\frac{32^{\frac{1}{3}}a^{5}c^{-\frac{5}{3}}}{\left(-27\right)^{\frac{1}{3}}\left(a^{6}\right)^{\frac{1}{3}}\left(c^{-3}\right)^{\frac{1}{3}}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and \frac{1}{3} to get 2.

$$\frac{32^{\frac{1}{3}}a^{5}c^{-\frac{5}{3}}}{\left(-27\right)^{\frac{1}{3}}a^{2}\left(c^{-3}\right)^{\frac{1}{3}}}$$

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

$$\frac{32^{\frac{1}{3}}a^{5}c^{-\frac{5}{3}}}{\left(-27\right)^{\frac{1}{3}}a^{2}c^{-1}}$$

Calculate -27 to the power of \frac{1}{3} and get -3.

$$\frac{32^{\frac{1}{3}}a^{5}c^{-\frac{5}{3}}}{-3a^{2}c^{-1}}$$

Cancel out a^{2} in both numerator and denominator.

$$\frac{\sqrt[3]{32}c^{-\frac{5}{3}}a^{3}}{-3\times \frac{1}{c}}$$

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

$$\frac{\sqrt[3]{32}a^{3}}{-3c^{\frac{2}{3}}}$$

Differentiate w.r.t. c

$$\frac{4\times 2^{\frac{2}{3}}a^{3}}{9c^{\frac{5}{3}}}$$