Solve (n^3)^3*(n^4)^5

$$( n ^ { 3 } ) ^ { 3 } \cdot ( n ^ { 4 } ) ^ { 5 }$$

Evaluate

$$n^{29}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(n^{3}\right)^{3}\left(n^{4}\right)^{5}$$

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

$$n^{3\times 3}n^{4\times 5}$$

Multiply 3 times 3.

$$n^{9}n^{4\times 5}$$

Multiply 4 times 5.

$$n^{9}n^{20}$$

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

$$n^{9+20}$$

Add the exponents 9 and 20.

$$n^{29}$$

Differentiate w.r.t. n

$$29n^{28}$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$\frac{\mathrm{d}}{\mathrm{d}n}(n^{9}\left(n^{4}\right)^{5})$$

To raise a power to another power, multiply the exponents. Multiply 4 and 5 to get 20.

$$\frac{\mathrm{d}}{\mathrm{d}n}(n^{9}n^{20})$$

To multiply powers of the same base, add their exponents. Add 9 and 20 to get 29.

$$\frac{\mathrm{d}}{\mathrm{d}n}(n^{29})$$

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

$$29n^{29-1}$$

Subtract 1 from 29.

$$29n^{28}$$