Solve frac{5-sqrt{2}}{2-sqrt{2}}=

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

Evaluate

$$\frac{3\sqrt{2}}{2}+4\approx 6.121320344$$

Solution Steps

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

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

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}.

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

Square 2. Square \sqrt{2}.

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

Subtract 2 from 4 to get 2.

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

Apply the distributive property by multiplying each term of 5-\sqrt{2} by each term of 2+\sqrt{2}.

$$\frac{10+5\sqrt{2}-2\sqrt{2}-\left(\sqrt{2}\right)^{2}}{2}$$

Combine 5\sqrt{2} and -2\sqrt{2} to get 3\sqrt{2}.

$$\frac{10+3\sqrt{2}-\left(\sqrt{2}\right)^{2}}{2}$$

The square of \sqrt{2} is 2.

$$\frac{10+3\sqrt{2}-2}{2}$$

Subtract 2 from 10 to get 8.

$$\frac{8+3\sqrt{2}}{2}$$