Solve (frac{x^{6}y^-9}{x^-3y^6})^1/3

$$( \frac { x ^ { 6 } y ^ { - 9 } } { x ^ { - 3 } y ^ { 6 } } ) ^ { 1 / 3 }$$

Evaluate

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

Solution Steps

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

$$\left(\frac{y^{-9}x^{9}}{y^{6}}\right)^{\frac{1}{3}}$$

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

$$\left(\frac{x^{9}}{y^{15}}\right)^{\frac{1}{3}}$$

To raise \frac{x^{9}}{y^{15}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(x^{9}\right)^{\frac{1}{3}}}{\left(y^{15}\right)^{\frac{1}{3}}}$$

To raise a power to another power, multiply the exponents. Multiply 9 and \frac{1}{3} to get 3.

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

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

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

Differentiate w.r.t. x

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