Solve (a^3)^2*(a^2)^5=

$$( a ^ { 3 } ) ^ { 2 } \cdot ( a ^ { 2 } ) ^ { 5 } =$$

Evaluate

$$a^{16}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

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

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

$$a^{3\times 2}a^{2\times 5}$$

Multiply 3 times 2.

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

Multiply 2 times 5.

$$a^{6}a^{10}$$

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

$$a^{6+10}$$

Add the exponents 6 and 10.

$$a^{16}$$

Differentiate w.r.t. a

$$16a^{15}$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{6}\left(a^{2}\right)^{5})$$

To raise a power to another power, multiply the exponents. Multiply 2 and 5 to get 10.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{6}a^{10})$$

To multiply powers of the same base, add their exponents. Add 6 and 10 to get 16.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{16})$$

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

$$16a^{16-1}$$

Subtract 1 from 16.

$$16a^{15}$$