Solve {left(x^3right)}^frac{1{3}}*{left(x^4right)}^frac{1{2}}*{left(x1/5right)}^5

$${ \left( { x }^{ 3 } \right) }^{ \frac{ 1 }{ 3 } } \times { \left( { x }^{ 4 } \right) }^{ \frac{ 1 }{ 2 } } \times { \left(x \frac{ 1 }{ 5 } \right) }^{ 5 }$$

Evaluate

$$\frac{x^{8}}{3125}$$

Solution Steps

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

$$x^{1}\left(x^{4}\right)^{\frac{1}{2}}\left(x\times \frac{1}{5}\right)^{5}$$

To raise a power to another power, multiply the exponents. Multiply 4 and \frac{1}{2} to get 2.

$$x^{1}x^{2}\left(x\times \frac{1}{5}\right)^{5}$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

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

Expand \left(x\times \frac{1}{5}\right)^{5}.

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

Calculate \frac{1}{5} to the power of 5 and get \frac{1}{3125}.

$$x^{3}x^{5}\times \frac{1}{3125}$$

To multiply powers of the same base, add their exponents. Add 3 and 5 to get 8.

$$x^{8}\times \frac{1}{3125}$$

Differentiate w.r.t. x

$$\frac{8x^{7}}{3125}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{1}\left(x^{4}\right)^{\frac{1}{2}}\left(x\times \frac{1}{5}\right)^{5})$$

To raise a power to another power, multiply the exponents. Multiply 4 and \frac{1}{2} to get 2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{1}x^{2}\left(x\times \frac{1}{5}\right)^{5})$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

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

Expand \left(x\times \frac{1}{5}\right)^{5}.

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

Calculate \frac{1}{5} to the power of 5 and get \frac{1}{3125}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}x^{5}\times \frac{1}{3125})$$

To multiply powers of the same base, add their exponents. Add 3 and 5 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{8}\times \frac{1}{3125})$$

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

$$8\times \frac{1}{3125}x^{8-1}$$

Multiply 8 times \frac{1}{3125}.

$$\frac{8}{3125}x^{8-1}$$

Subtract 1 from 8.

$$\frac{8}{3125}x^{7}$$