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

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

Evaluate

$$2a^{19}$$

Solution Steps

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

$$2a^{15}\left(a^{2}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$2a^{15}a^{4}$$

To multiply powers of the same base, add their exponents. Add 15 and 4 to get 19.

$$2a^{19}$$

Differentiate w.r.t. a

$$38a^{18}$$

Steps Using Definition of a Derivative

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

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{\mathrm{d}}{\mathrm{d}a}(2a^{15}a^{4})$$

To multiply powers of the same base, add their exponents. Add 15 and 4 to get 19.

$$\frac{\mathrm{d}}{\mathrm{d}a}(2a^{19})$$

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

$$19\times 2a^{19-1}$$

Multiply 19 times 2.

$$38a^{19-1}$$

Subtract 1 from 19.

$$38a^{18}$$