Solve [(x^-2)/(xy)^3]^1.5

$$[ \frac { ( x ^ { - 2 } ) } { ( x y ) ^ { 3 } } ] ^ { 1.5 }$$

Evaluate

$$\frac{1}{\left(yx^{\frac{5}{3}}\right)^{4.5}}$$

Solution Steps

Expand \left(xy\right)^{3}.

$$\left(\frac{x^{-2}}{x^{3}y^{3}}\right)^{1.5}$$

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

$$\left(\frac{1}{y^{3}x^{5}}\right)^{1.5}$$

To raise \frac{1}{y^{3}x^{5}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{1^{1.5}}{\left(y^{3}x^{5}\right)^{1.5}}$$

Calculate 1 to the power of 1.5 and get 1.

$$\frac{1}{\left(y^{3}x^{5}\right)^{1.5}}$$

Expand \left(y^{3}x^{5}\right)^{1.5}.

$$\frac{1}{\left(y^{3}\right)^{1.5}\left(x^{5}\right)^{1.5}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 1.5 to get 4.5.

$$\frac{1}{y^{4.5}\left(x^{5}\right)^{1.5}}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 1.5 to get 7.5.

$$\frac{1}{y^{4.5}x^{7.5}}$$

Differentiate w.r.t. x

$$-\frac{15y}{2\left(yx^{\frac{17}{11}}\right)^{5.5}}$$