Solve (w^3)^8*(w^5)^5

$$( w ^ { 3 } ) ^ { 8 } \cdot ( w ^ { 5 } ) ^ { 5 }$$

Evaluate

$$w^{49}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(w^{3}\right)^{8}\left(w^{5}\right)^{5}$$

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

$$w^{3\times 8}w^{5\times 5}$$

Multiply 3 times 8.

$$w^{24}w^{5\times 5}$$

Multiply 5 times 5.

$$w^{24}w^{25}$$

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

$$w^{24+25}$$

Add the exponents 24 and 25.

$$w^{49}$$

Differentiate w.r.t. w

$$49w^{48}$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply 3 and 8 to get 24.

$$\frac{\mathrm{d}}{\mathrm{d}w}(w^{24}\left(w^{5}\right)^{5})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}w}(w^{24}w^{25})$$

To multiply powers of the same base, add their exponents. Add 24 and 25 to get 49.

$$\frac{\mathrm{d}}{\mathrm{d}w}(w^{49})$$

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

$$49w^{49-1}$$

Subtract 1 from 49.

$$49w^{48}$$