Solve 1/5-sqrt{2}

$$\frac { 1 } { 5 - \sqrt { 2 } }$$

Evaluate

$$\frac{\sqrt{2}+5}{23}\approx 0.278878851$$

Solution Steps

Rationalize the denominator of \frac{1}{5-\sqrt{2}} by multiplying numerator and denominator by 5+\sqrt{2}.

$$\frac{5+\sqrt{2}}{\left(5-\sqrt{2}\right)\left(5+\sqrt{2}\right)}$$

Consider \left(5-\sqrt{2}\right)\left(5+\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}.

$$\frac{5+\sqrt{2}}{5^{2}-\left(\sqrt{2}\right)^{2}}$$

Square 5. Square \sqrt{2}.

$$\frac{5+\sqrt{2}}{25-2}$$

Subtract 2 from 25 to get 23.

$$\frac{5+\sqrt{2}}{23}$$