Solve (3p^4)^-2*3p^0

$$( 3 p ^ { 4 } ) ^ { - 2 } \times 3 p ^ { 0 }$$

Evaluate

$$\frac{1}{3p^{8}}$$

Solution Steps

Expand \left(3p^{4}\right)^{-2}.

$$3^{-2}\left(p^{4}\right)^{-2}\times 3p^{0}$$

To raise a power to another power, multiply the exponents. Multiply 4 and -2 to get -8.

$$3^{-2}p^{-8}\times 3p^{0}$$

Calculate 3 to the power of -2 and get \frac{1}{9}.

$$\frac{1}{9}p^{-8}\times 3p^{0}$$

Multiply \frac{1}{9} and 3 to get \frac{1}{3}.

$$\frac{1}{3}p^{-8}p^{0}$$

Calculate p to the power of 0 and get 1.

$$\frac{1}{3}p^{-8}\times 1$$

Multiply \frac{1}{3} and 1 to get \frac{1}{3}.

$$\frac{1}{3}p^{-8}$$

Differentiate w.r.t. p

$$-\frac{8}{3p^{9}}$$

Steps Using Definition of a Derivative

Expand \left(3p^{4}\right)^{-2}.

$$\frac{\mathrm{d}}{\mathrm{d}p}(3^{-2}\left(p^{4}\right)^{-2}\times 3p^{0})$$

To raise a power to another power, multiply the exponents. Multiply 4 and -2 to get -8.

$$\frac{\mathrm{d}}{\mathrm{d}p}(3^{-2}p^{-8}\times 3p^{0})$$

Calculate 3 to the power of -2 and get \frac{1}{9}.

$$\frac{\mathrm{d}}{\mathrm{d}p}(\frac{1}{9}p^{-8}\times 3p^{0})$$

Multiply \frac{1}{9} and 3 to get \frac{1}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}p}(\frac{1}{3}p^{-8}p^{0})$$

Calculate p to the power of 0 and get 1.

$$\frac{\mathrm{d}}{\mathrm{d}p}(\frac{1}{3}p^{-8}\times 1)$$

Multiply \frac{1}{3} and 1 to get \frac{1}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}p}(\frac{1}{3}p^{-8})$$

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

$$-8\times \frac{1}{3}p^{-8-1}$$

Multiply -8 times \frac{1}{3}.

$$-\frac{8}{3}p^{-8-1}$$

Subtract 1 from -8.

$$-\frac{8}{3}p^{-9}$$