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

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

Evaluate

$$x^{14}$$

Solution Steps

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

$$x^{6}x^{2^{4}}x^{-2^{3}}$$

Calculate 2 to the power of 4 and get 16.

$$x^{6}x^{16}x^{-2^{3}}$$

To multiply powers of the same base, add their exponents. Add 6 and 16 to get 22.

$$x^{22}x^{-2^{3}}$$

Calculate 2 to the power of 3 and get 8.

$$x^{22}x^{-8}$$

To multiply powers of the same base, add their exponents. Add 22 and -8 to get 14.

$$x^{14}$$

Differentiate w.r.t. x

$$14x^{13}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{6}x^{2^{4}}x^{-2^{3}})$$

Calculate 2 to the power of 4 and get 16.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{6}x^{16}x^{-2^{3}})$$

To multiply powers of the same base, add their exponents. Add 6 and 16 to get 22.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{22}x^{-2^{3}})$$

Calculate 2 to the power of 3 and get 8.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{22}x^{-8})$$

To multiply powers of the same base, add their exponents. Add 22 and -8 to get 14.

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

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

$$14x^{14-1}$$

Subtract 1 from 14.

$$14x^{13}$$