Solve (16x^2/3y^-2)^-3/2/2x^-2y

$$\frac { ( 16 x ^ { 2 / 3 } y ^ { - 2 } ) ^ { - 3 / 2 } } { 2 x ^ { - 2 } y }$$

Evaluate

$$\frac{x^{2}\left(|y|\right)^{3}}{128y|x|}$$

Solution Steps

Expand \left(16x^{\frac{2}{3}}y^{-2}\right)^{-\frac{3}{2}}.

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

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

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

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

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

Calculate 16 to the power of -\frac{3}{2} and get \frac{1}{64}.

$$\frac{\frac{1}{64}x^{-1}y^{3}}{2x^{-2}y}$$

Cancel out y in both numerator and denominator.

$$\frac{\frac{1}{64}\times \frac{1}{x}y^{2}}{2x^{-2}}$$

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

$$\frac{\frac{1}{64}x^{1}y^{2}}{2}$$

Calculate x to the power of 1 and get x.

$$\frac{\frac{1}{64}xy^{2}}{2}$$

Divide \frac{1}{64}xy^{2} by 2 to get \frac{1}{128}xy^{2}.

$$\frac{1}{128}xy^{2}$$

Differentiate w.r.t. x

$$\frac{x\left(|y|\right)^{3}}{128y|x|}$$