Solve x^{3}(x^2)^3x^0x^-2

$$x ^ { 3 } ( x ^ { 2 } ) ^ { 3 } x ^ { 0 } x ^ { - 2 }$$

Evaluate

$$x^{7}$$

Solution Steps

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

$$x^{3}x^{6}x^{0}x^{-2}$$

To multiply powers of the same base, add their exponents. Add 3 and 6 to get 9.

$$x^{9}x^{0}x^{-2}$$

To multiply powers of the same base, add their exponents. Add 9 and 0 to get 9.

$$x^{9}x^{-2}$$

To multiply powers of the same base, add their exponents. Add 9 and -2 to get 7.

$$x^{7}$$

Differentiate w.r.t. x

$$7x^{6}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}x^{6}x^{0}x^{-2})$$

To multiply powers of the same base, add their exponents. Add 3 and 6 to get 9.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{9}x^{0}x^{-2})$$

To multiply powers of the same base, add their exponents. Add 9 and 0 to get 9.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{9}x^{-2})$$

To multiply powers of the same base, add their exponents. Add 9 and -2 to get 7.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{7})$$

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

$$7x^{7-1}$$

Subtract 1 from 7.

$$7x^{6}$$