Solve (a^5)^3*(a^2)^6

$$( a ^ { 5 } ) ^ { 3 } \cdot ( a ^ { 2 } ) ^ { 6 }$$

Evaluate

$$a^{27}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(a^{5}\right)^{3}\left(a^{2}\right)^{6}$$

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

$$a^{5\times 3}a^{2\times 6}$$

Multiply 5 times 3.

$$a^{15}a^{2\times 6}$$

Multiply 2 times 6.

$$a^{15}a^{12}$$

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

$$a^{15+12}$$

Add the exponents 15 and 12.

$$a^{27}$$

Differentiate w.r.t. a

$$27a^{26}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{15}\left(a^{2}\right)^{6})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{15}a^{12})$$

To multiply powers of the same base, add their exponents. Add 15 and 12 to get 27.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{27})$$

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

$$27a^{27-1}$$

Subtract 1 from 27.

$$27a^{26}$$