Solve p^{frac{1/5}p^frac{7{10}}p^frac{1{2}}}{(p^3)^-frac{1{5}}}

$$\frac { p ^ { \frac { 1 } { 5 } } p ^ { \frac { 7 } { 10 } } p ^ { \frac { 1 } { 2 } } } { ( p ^ { 3 } ) ^ { - \frac { 1 } { 5 } } }$$

Evaluate

$$p^{2}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add \frac{1}{5} and \frac{7}{10} to get \frac{9}{10}.

$$\frac{p^{\frac{9}{10}}p^{\frac{1}{2}}}{\left(p^{3}\right)^{-\frac{1}{5}}}$$

To multiply powers of the same base, add their exponents. Add \frac{9}{10} and \frac{1}{2} to get \frac{7}{5}.

$$\frac{p^{\frac{7}{5}}}{\left(p^{3}\right)^{-\frac{1}{5}}}$$

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

$$\frac{p^{\frac{7}{5}}}{p^{-\frac{3}{5}}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract -\frac{3}{5} from \frac{7}{5} to get 2.

$$p^{2}$$

Differentiate w.r.t. p

$$2p$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add \frac{1}{5} and \frac{7}{10} to get \frac{9}{10}.

$$\frac{\mathrm{d}}{\mathrm{d}p}(\frac{p^{\frac{9}{10}}p^{\frac{1}{2}}}{\left(p^{3}\right)^{-\frac{1}{5}}})$$

To multiply powers of the same base, add their exponents. Add \frac{9}{10} and \frac{1}{2} to get \frac{7}{5}.

$$\frac{\mathrm{d}}{\mathrm{d}p}(\frac{p^{\frac{7}{5}}}{\left(p^{3}\right)^{-\frac{1}{5}}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}p}(\frac{p^{\frac{7}{5}}}{p^{-\frac{3}{5}}})$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract -\frac{3}{5} from \frac{7}{5} to get 2.

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

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

$$2p^{2-1}$$

Subtract 1 from 2.

$$2p^{1}$$

For any term t, t^{1}=t.

$$2p$$