Solve (w^3)^5(w^2)^4

$$( w ^ { 3 } ) ^ { 5 } ( w ^ { 2 } ) ^ { 4 }$$

Evaluate

$$w^{23}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(w^{3}\right)^{5}\left(w^{2}\right)^{4}$$

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

$$w^{3\times 5}w^{2\times 4}$$

Multiply 3 times 5.

$$w^{15}w^{2\times 4}$$

Multiply 2 times 4.

$$w^{15}w^{8}$$

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

$$w^{15+8}$$

Add the exponents 15 and 8.

$$w^{23}$$

Differentiate w.r.t. w

$$23w^{22}$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply 3 and 5 to get 15.

$$\frac{\mathrm{d}}{\mathrm{d}w}(w^{15}\left(w^{2}\right)^{4})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}w}(w^{15}w^{8})$$

To multiply powers of the same base, add their exponents. Add 15 and 8 to get 23.

$$\frac{\mathrm{d}}{\mathrm{d}w}(w^{23})$$

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

$$23w^{23-1}$$

Subtract 1 from 23.

$$23w^{22}$$