Solve a*left(a^2right)^2*left(a^3right)^3

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

Evaluate

$$a^{14}$$

Solution Steps

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

$$aa^{4}\left(a^{3}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$aa^{4}a^{9}$$

To multiply powers of the same base, add their exponents. Add 1 and 4 to get 5.

$$a^{5}a^{9}$$

To multiply powers of the same base, add their exponents. Add 5 and 9 to get 14.

$$a^{14}$$

Differentiate w.r.t. a

$$14a^{13}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(aa^{4}\left(a^{3}\right)^{3})$$

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$\frac{\mathrm{d}}{\mathrm{d}a}(aa^{4}a^{9})$$

To multiply powers of the same base, add their exponents. Add 1 and 4 to get 5.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{5}a^{9})$$

To multiply powers of the same base, add their exponents. Add 5 and 9 to get 14.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{14})$$

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

$$14a^{14-1}$$

Subtract 1 from 14.

$$14a^{13}$$