Solve (8y^12/27x^15)^frac{1{3}}

$$( \frac { 8 y ^ { 12 } } { 27 x ^ { 15 } } ) ^ { \frac { 1 } { 3 } }$$

Evaluate

$$\frac{2y^{4}}{3x^{5}}$$

Solution Steps

To raise \frac{8y^{12}}{27x^{15}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(8y^{12}\right)^{\frac{1}{3}}}{\left(27x^{15}\right)^{\frac{1}{3}}}$$

Expand \left(8y^{12}\right)^{\frac{1}{3}}.

$$\frac{8^{\frac{1}{3}}\left(y^{12}\right)^{\frac{1}{3}}}{\left(27x^{15}\right)^{\frac{1}{3}}}$$

To raise a power to another power, multiply the exponents. Multiply 12 and \frac{1}{3} to get 4.

$$\frac{8^{\frac{1}{3}}y^{4}}{\left(27x^{15}\right)^{\frac{1}{3}}}$$

Calculate 8 to the power of \frac{1}{3} and get 2.

$$\frac{2y^{4}}{\left(27x^{15}\right)^{\frac{1}{3}}}$$

Expand \left(27x^{15}\right)^{\frac{1}{3}}.

$$\frac{2y^{4}}{27^{\frac{1}{3}}\left(x^{15}\right)^{\frac{1}{3}}}$$

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

$$\frac{2y^{4}}{27^{\frac{1}{3}}x^{5}}$$

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

$$\frac{2y^{4}}{3x^{5}}$$

Differentiate w.r.t. x

$$-\frac{10y^{4}}{3x^{6}}$$