Solve 2(a^{5})^{2}*(a^{2})^2-(a^2)^4*a^6

$$2 ( a ^ { 5 } ) ^ { 2 } \times ( a ^ { 2 } ) ^ { 2 } - ( a ^ { 2 } ) ^ { 4 } \times a ^ { 6 }$$

Evaluate

$$a^{14}$$

Solution Steps

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

$$2a^{10}\left(a^{2}\right)^{2}-\left(a^{2}\right)^{4}a^{6}$$

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

$$2a^{10}a^{4}-\left(a^{2}\right)^{4}a^{6}$$

To multiply powers of the same base, add their exponents. Add 10 and 4 to get 14.

$$2a^{14}-\left(a^{2}\right)^{4}a^{6}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 4 to get 8.

$$2a^{14}-a^{8}a^{6}$$

To multiply powers of the same base, add their exponents. Add 8 and 6 to get 14.

$$2a^{14}-a^{14}$$

Combine 2a^{14} and -a^{14} to get a^{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 5 and 2 to get 10.

$$\frac{\mathrm{d}}{\mathrm{d}a}(2a^{10}\left(a^{2}\right)^{2}-\left(a^{2}\right)^{4}a^{6})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(2a^{10}a^{4}-\left(a^{2}\right)^{4}a^{6})$$

To multiply powers of the same base, add their exponents. Add 10 and 4 to get 14.

$$\frac{\mathrm{d}}{\mathrm{d}a}(2a^{14}-\left(a^{2}\right)^{4}a^{6})$$

To raise a power to another power, multiply the exponents. Multiply 2 and 4 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}a}(2a^{14}-a^{8}a^{6})$$

To multiply powers of the same base, add their exponents. Add 8 and 6 to get 14.

$$\frac{\mathrm{d}}{\mathrm{d}a}(2a^{14}-a^{14})$$

Combine 2a^{14} and -a^{14} to get a^{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.

$$14a^{13}$$