Solve ffrac{(4+sqrt{2})(4-sqrt{2})}{sqrt{7}}

$$f \frac { ( 4 + \sqrt { 2 } ) ( 4 - \sqrt { 2 } ) } { \sqrt { 7 } }$$

Evaluate

$$2\sqrt{7}f$$

Solution Steps

Consider \left(4+\sqrt{2}\right)\left(4-\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}.

$$f\times \frac{4^{2}-\left(\sqrt{2}\right)^{2}}{\sqrt{7}}$$

Calculate 4 to the power of 2 and get 16.

$$f\times \frac{16-\left(\sqrt{2}\right)^{2}}{\sqrt{7}}$$

The square of \sqrt{2} is 2.

$$f\times \frac{16-2}{\sqrt{7}}$$

Subtract 2 from 16 to get 14.

$$f\times \frac{14}{\sqrt{7}}$$

Rationalize the denominator of \frac{14}{\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$f\times \frac{14\sqrt{7}}{\left(\sqrt{7}\right)^{2}}$$

The square of \sqrt{7} is 7.

$$f\times \frac{14\sqrt{7}}{7}$$

Divide 14\sqrt{7} by 7 to get 2\sqrt{7}.

$$f\times 2\sqrt{7}$$

Differentiate w.r.t. f

$$2 \sqrt{7} = 5.291502622$$