Solve (2^4*4)^frac{1{2}}div(p^frac{1{3}})^3=

$$( 2 ^ { 4 } \times 4 ) ^ { \frac { 1 } { 2 } } \div ( p ^ { \frac { 1 } { 3 } } ) ^ { 3 } =$$

Evaluate

$$\frac{8}{p}$$

Solution Steps

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

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

Calculate 2 to the power of 4 and get 16.

$$\frac{\left(16\times 4\right)^{\frac{1}{2}}}{p^{1}}$$

Multiply 16 and 4 to get 64.

$$\frac{64^{\frac{1}{2}}}{p^{1}}$$

Calculate 64 to the power of \frac{1}{2} and get 8.

$$\frac{8}{p^{1}}$$

Calculate p to the power of 1 and get p.

$$\frac{8}{p}$$

Differentiate w.r.t. p

$$-\frac{8}{p^{2}}$$

Steps Using Definition of a Derivative

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

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

Calculate 2 to the power of 4 and get 16.

$$\frac{\mathrm{d}}{\mathrm{d}p}(\frac{\left(16\times 4\right)^{\frac{1}{2}}}{p^{1}})$$

Multiply 16 and 4 to get 64.

$$\frac{\mathrm{d}}{\mathrm{d}p}(\frac{64^{\frac{1}{2}}}{p^{1}})$$

Calculate 64 to the power of \frac{1}{2} and get 8.

$$\frac{\mathrm{d}}{\mathrm{d}p}(\frac{8}{p^{1}})$$

Calculate p to the power of 1 and get p.

$$\frac{\mathrm{d}}{\mathrm{d}p}(\frac{8}{p})$$

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

$$-8p^{-1-1}$$

Subtract 1 from -1.

$$-8p^{-2}$$