Solve =(n^5)^5/(n^-7)

$$= \frac { ( n ^ { 5 } ) ^ { 5 } } { ( n ^ { - 7 } ) }$$

Evaluate

$$n^{32}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(n^{5}\right)^{5}\times \frac{1}{n^{-7}}$$

To raise a power to another power, multiply the exponents.

$$n^{5\times 5}n^{-7\left(-1\right)}$$

Multiply 5 times 5.

$$n^{25}n^{-7\left(-1\right)}$$

Multiply -7 times -1.

$$n^{25}n^{7}$$

To multiply powers of the same base, add their exponents.

$$n^{25+7}$$

Add the exponents 25 and 7.

$$n^{32}$$

Differentiate w.r.t. n

$$32n^{31}$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply 5 and 5 to get 25.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{n^{25}}{n^{-7}})$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract -7 from 25 to get 32.

$$\frac{\mathrm{d}}{\mathrm{d}n}(n^{32})$$

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

$$32n^{32-1}$$

Subtract 1 from 32.

$$32n^{31}$$