Solve (x^8y^-4/16y^frac{4{3}})^-1/4

$$( \frac { x ^ { 8 } y ^ { - 4 } } { 16 y ^ { \frac { 4 } { 3 } } } ) ^ { - 1 / 4 }$$

Evaluate

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

Solution Steps

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

$$\left(\frac{x^{8}}{16y^{\frac{16}{3}}}\right)^{-\frac{1}{4}}$$

To raise \frac{x^{8}}{16y^{\frac{16}{3}}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(x^{8}\right)^{-\frac{1}{4}}}{\left(16y^{\frac{16}{3}}\right)^{-\frac{1}{4}}}$$

To raise a power to another power, multiply the exponents. Multiply 8 and -\frac{1}{4} to get -2.

$$\frac{x^{-2}}{\left(16y^{\frac{16}{3}}\right)^{-\frac{1}{4}}}$$

Expand \left(16y^{\frac{16}{3}}\right)^{-\frac{1}{4}}.

$$\frac{x^{-2}}{16^{-\frac{1}{4}}\left(y^{\frac{16}{3}}\right)^{-\frac{1}{4}}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{16}{3} and -\frac{1}{4} to get -\frac{4}{3}.

$$\frac{x^{-2}}{16^{-\frac{1}{4}}y^{-\frac{4}{3}}}$$

Calculate 16 to the power of -\frac{1}{4} and get \frac{1}{2}.

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

Differentiate w.r.t. x

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