Solve {left(sqrt{x^4}right)}^2*{left(sqrt{x^8}right)}^2*{left(sqrt{x^2}right)}^2

$${ \left( \sqrt{ { x }^{ 4 } } \right) }^{ 2 } \times { \left( \sqrt{ { x }^{ 8 } } \right) }^{ 2 } \times { \left( \sqrt{ { x }^{ 2 } } \right) }^{ 2 }$$

Evaluate

$$x^{14}$$

Solution Steps

Calculate \sqrt{x^{4}} to the power of 2 and get x^{4}.

$$x^{4}\left(\sqrt{x^{8}}\right)^{2}\left(\sqrt{x^{2}}\right)^{2}$$

Calculate \sqrt{x^{8}} to the power of 2 and get x^{8}.

$$x^{4}x^{8}\left(\sqrt{x^{2}}\right)^{2}$$

To multiply powers of the same base, add their exponents. Add 4 and 8 to get 12.

$$x^{12}\left(\sqrt{x^{2}}\right)^{2}$$

Calculate \sqrt{x^{2}} to the power of 2 and get x^{2}.

$$x^{12}x^{2}$$

To multiply powers of the same base, add their exponents. Add 12 and 2 to get 14.

$$x^{14}$$

Differentiate w.r.t. x

$$14x^{13}$$

Steps Using Definition of a Derivative

Calculate \sqrt{x^{4}} to the power of 2 and get x^{4}.

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

Calculate \sqrt{x^{8}} to the power of 2 and get x^{8}.

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

To multiply powers of the same base, add their exponents. Add 4 and 8 to get 12.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{12}\left(\sqrt{x^{2}}\right)^{2})$$

Calculate \sqrt{x^{2}} to the power of 2 and get x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{12}x^{2})$$

To multiply powers of the same base, add their exponents. Add 12 and 2 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.

$$14x^{13}$$