Solve [(x^32)^frac{1{3}}]^frac{1{5}}

$$[ ( x ^ { 32 } ) ^ { \frac { 1 } { 3 } } ] ^ { \frac { 1 } { 5 } }$$

Differentiate w.r.t. x

$$\frac{32x^{\frac{17}{15}}}{15}$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply 32 and \frac{1}{3} to get \frac{32}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(x^{\frac{32}{3}}\right)^{\frac{1}{5}})$$

To raise a power to another power, multiply the exponents. Multiply \frac{32}{3} and \frac{1}{5} to get \frac{32}{15}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{\frac{32}{15}})$$

The derivative of ax^{n} is nax^{n-1}.

$$\frac{32}{15}x^{\frac{32}{15}-1}$$

Subtract 1 from \frac{32}{15}.

$$\frac{32}{15}x^{\frac{17}{15}}$$

Evaluate

$$x^{\frac{32}{15}}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 32 and \frac{1}{3} to get \frac{32}{3}.

$$\left(x^{\frac{32}{3}}\right)^{\frac{1}{5}}$$