Solve (x^2y)^frac{5{3}}(x^-1)^frac{1{2}}

$$( x ^ { 2 } y ) ^ { \frac { 5 } { 3 } } ( x ^ { - 1 } ) ^ { \frac { 1 } { 2 } }$$

Evaluate

$$y^{\frac{5}{3}}x^{\frac{17}{6}}$$

Solution Steps

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

$$\left(x^{2}y\right)^{\frac{5}{3}}x^{-\frac{1}{2}}$$

Expand \left(x^{2}y\right)^{\frac{5}{3}}.

$$\left(x^{2}\right)^{\frac{5}{3}}y^{\frac{5}{3}}x^{-\frac{1}{2}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and \frac{5}{3} to get \frac{10}{3}.

$$x^{\frac{10}{3}}y^{\frac{5}{3}}x^{-\frac{1}{2}}$$

To multiply powers of the same base, add their exponents. Add \frac{10}{3} and -\frac{1}{2} to get \frac{17}{6}.

$$x^{\frac{17}{6}}y^{\frac{5}{3}}$$

Differentiate w.r.t. x

$$\frac{17y^{\frac{5}{3}}x^{\frac{11}{6}}}{6}$$