Solve (9x^-2/16y^-4)^-3/2

$$( \frac { 9 x ^ { - 2 } } { 16 y ^ { - 4 } } ) ^ { - 3 / 2 }$$

Evaluate

$$\frac{64\left(|x|\right)^{3}}{27y^{6}}$$

Solution Steps

To raise \frac{9x^{-2}}{16y^{-4}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(9x^{-2}\right)^{-\frac{3}{2}}}{\left(16y^{-4}\right)^{-\frac{3}{2}}}$$

Expand \left(9x^{-2}\right)^{-\frac{3}{2}}.

$$\frac{9^{-\frac{3}{2}}\left(x^{-2}\right)^{-\frac{3}{2}}}{\left(16y^{-4}\right)^{-\frac{3}{2}}}$$

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

$$\frac{9^{-\frac{3}{2}}x^{3}}{\left(16y^{-4}\right)^{-\frac{3}{2}}}$$

Calculate 9 to the power of -\frac{3}{2} and get \frac{1}{27}.

$$\frac{\frac{1}{27}x^{3}}{\left(16y^{-4}\right)^{-\frac{3}{2}}}$$

Expand \left(16y^{-4}\right)^{-\frac{3}{2}}.

$$\frac{\frac{1}{27}x^{3}}{16^{-\frac{3}{2}}\left(y^{-4}\right)^{-\frac{3}{2}}}$$

To raise a power to another power, multiply the exponents. Multiply -4 and -\frac{3}{2} to get 6.

$$\frac{\frac{1}{27}x^{3}}{16^{-\frac{3}{2}}y^{6}}$$

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

$$\frac{\frac{1}{27}x^{3}}{\frac{1}{64}y^{6}}$$

Differentiate w.r.t. x

$$\frac{64x|x|}{9y^{6}}$$