Solve (n-2sqrt{2})(n+2sqrt{2})

$$( n - 2 \sqrt { 2 } ) ( n + 2 \sqrt { 2 } )$$

Evaluate

$$n^{2}-8$$

Solution Steps

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$n^{2}-\left(2\sqrt{2}\right)^{2}$$

Expand \left(2\sqrt{2}\right)^{2}.

$$n^{2}-2^{2}\left(\sqrt{2}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$n^{2}-4\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$n^{2}-4\times 2$$

Multiply 4 and 2 to get 8.

$$n^{2}-8$$

Differentiate w.r.t. n

$$2n$$

Steps Using Derivative Rule for Sum

Consider \left(n-2\sqrt{2}\right)\left(n+2\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}n}(n^{2}-\left(2\sqrt{2}\right)^{2})$$

Expand \left(2\sqrt{2}\right)^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}n}(n^{2}-2^{2}\left(\sqrt{2}\right)^{2})$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\mathrm{d}}{\mathrm{d}n}(n^{2}-4\left(\sqrt{2}\right)^{2})$$

The square of \sqrt{2} is 2.

$$\frac{\mathrm{d}}{\mathrm{d}n}(n^{2}-4\times 2)$$

Multiply 4 and 2 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}n}(n^{2}-8)$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$2n^{2-1}$$

Subtract 1 from 2.

$$2n^{1}$$

For any term t, t^{1}=t.

$$2n$$