Solve left(A^2right)^6left(A^3right)^8

$${ \left( { A }^{ 2 } \right) }^{ 6 } { \left( { A }^{ 3 } \right) }^{ 8 }$$

Evaluate

$$A^{36}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(A^{2}\right)^{6}\left(A^{3}\right)^{8}$$

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

$$A^{2\times 6}A^{3\times 8}$$

Multiply 2 times 6.

$$A^{12}A^{3\times 8}$$

Multiply 3 times 8.

$$A^{12}A^{24}$$

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

$$A^{12+24}$$

Add the exponents 12 and 24.

$$A^{36}$$

Differentiate w.r.t. A

$$36A^{35}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}A}(A^{12}\left(A^{3}\right)^{8})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}A}(A^{12}A^{24})$$

To multiply powers of the same base, add their exponents. Add 12 and 24 to get 36.

$$\frac{\mathrm{d}}{\mathrm{d}A}(A^{36})$$

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

$$36A^{36-1}$$

Subtract 1 from 36.

$$36A^{35}$$