Solve 5/2-sqrt{2}

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

Evaluate

$$\frac{5\sqrt{2}}{2}+5\approx 8.535533906$$

Solution Steps

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

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

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

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

Square 2. Square \sqrt{2}.

$$\frac{5\left(2+\sqrt{2}\right)}{4-2}$$

Subtract 2 from 4 to get 2.

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

Use the distributive property to multiply 5 by 2+\sqrt{2}.

$$\frac{10+5\sqrt{2}}{2}$$