Solve (x^frac{1{2}}y^-frac{2{4}})^-2^3=

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

Evaluate

$$\left(\frac{y}{x}\right)^{4}$$

Solution Steps

Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.

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

Calculate 2 to the power of 3 and get 8.

$$\left(x^{\frac{1}{2}}y^{-\frac{1}{2}}\right)^{-8}$$

Expand \left(x^{\frac{1}{2}}y^{-\frac{1}{2}}\right)^{-8}.

$$\left(x^{\frac{1}{2}}\right)^{-8}\left(y^{-\frac{1}{2}}\right)^{-8}$$

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

$$x^{-4}\left(y^{-\frac{1}{2}}\right)^{-8}$$

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

$$x^{-4}y^{4}$$

Differentiate w.r.t. x

$$-\frac{4y^{4}}{x^{5}}$$