Solve [4x^-2/9y^-4]^-frac{5{2}}

$$[ \frac { 4 x ^ { - 2 } } { 9 y ^ { - 4 } } ] ^ { - \frac { 5 } { 2 } }$$

Evaluate

$$\frac{243\left(|x|\right)^{5}}{32y^{10}}$$

Solution Steps

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

$$\frac{\left(4x^{-2}\right)^{-\frac{5}{2}}}{\left(9y^{-4}\right)^{-\frac{5}{2}}}$$

Expand \left(4x^{-2}\right)^{-\frac{5}{2}}.

$$\frac{4^{-\frac{5}{2}}\left(x^{-2}\right)^{-\frac{5}{2}}}{\left(9y^{-4}\right)^{-\frac{5}{2}}}$$

To raise a power to another power, multiply the exponents. Multiply -2 and -\frac{5}{2} to get 5.

$$\frac{4^{-\frac{5}{2}}x^{5}}{\left(9y^{-4}\right)^{-\frac{5}{2}}}$$

Calculate 4 to the power of -\frac{5}{2} and get \frac{1}{32}.

$$\frac{\frac{1}{32}x^{5}}{\left(9y^{-4}\right)^{-\frac{5}{2}}}$$

Expand \left(9y^{-4}\right)^{-\frac{5}{2}}.

$$\frac{\frac{1}{32}x^{5}}{9^{-\frac{5}{2}}\left(y^{-4}\right)^{-\frac{5}{2}}}$$

To raise a power to another power, multiply the exponents. Multiply -4 and -\frac{5}{2} to get 10.

$$\frac{\frac{1}{32}x^{5}}{9^{-\frac{5}{2}}y^{10}}$$

Calculate 9 to the power of -\frac{5}{2} and get \frac{1}{243}.

$$\frac{\frac{1}{32}x^{5}}{\frac{1}{243}y^{10}}$$

Differentiate w.r.t. x

$$\frac{1215|x|x^{3}}{32y^{10}}$$