Solve (x^{2})^{3}*x^{2}-(x^4)^2+x^2*x^6

$$( x ^ { 2 } ) ^ { 3 } \cdot x ^ { 2 } - ( x ^ { 4 } ) ^ { 2 } + x ^ { 2 } \cdot x ^ { 6 }$$

Evaluate

$$x^{8}$$

Solution Steps

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

$$x^{6}x^{2}-\left(x^{4}\right)^{2}+x^{2}x^{6}$$

To multiply powers of the same base, add their exponents. Add 6 and 2 to get 8.

$$x^{8}-\left(x^{4}\right)^{2}+x^{2}x^{6}$$

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

$$x^{8}-x^{8}+x^{2}x^{6}$$

To multiply powers of the same base, add their exponents. Add 2 and 6 to get 8.

$$x^{8}-x^{8}+x^{8}$$

Combine x^{8} and -x^{8} to get 0.

$$0+x^{8}$$

Anything plus zero gives itself.

$$x^{8}$$

Differentiate w.r.t. x

$$8x^{7}$$

Steps Using Definition of a Derivative

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

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

To multiply powers of the same base, add their exponents. Add 6 and 2 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{8}-\left(x^{4}\right)^{2}+x^{2}x^{6})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{8}-x^{8}+x^{2}x^{6})$$

To multiply powers of the same base, add their exponents. Add 2 and 6 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{8}-x^{8}+x^{8})$$

Combine x^{8} and -x^{8} to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}x}(0+x^{8})$$

Anything plus zero gives itself.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{8})$$

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

$$8x^{8-1}$$

Subtract 1 from 8.

$$8x^{7}$$