Solve (u^2)^6(u^5)^6

$$( u ^ { 2 } ) ^ { 6 } ( u ^ { 5 } ) ^ { 6 }$$

Evaluate

$$u^{42}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(u^{2}\right)^{6}\left(u^{5}\right)^{6}$$

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

$$u^{2\times 6}u^{5\times 6}$$

Multiply 2 times 6.

$$u^{12}u^{5\times 6}$$

Multiply 5 times 6.

$$u^{12}u^{30}$$

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

$$u^{12+30}$$

Add the exponents 12 and 30.

$$u^{42}$$

Differentiate w.r.t. u

$$42u^{41}$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply 2 and 6 to get 12.

$$\frac{\mathrm{d}}{\mathrm{d}u}(u^{12}\left(u^{5}\right)^{6})$$

To raise a power to another power, multiply the exponents. Multiply 5 and 6 to get 30.

$$\frac{\mathrm{d}}{\mathrm{d}u}(u^{12}u^{30})$$

To multiply powers of the same base, add their exponents. Add 12 and 30 to get 42.

$$\frac{\mathrm{d}}{\mathrm{d}u}(u^{42})$$

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

$$42u^{42-1}$$

Subtract 1 from 42.

$$42u^{41}$$