Solve frac{5+2sqrt{2}}{2+sqrt{2}}

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

Evaluate

$$-\frac{\sqrt{2}}{2}+3\approx 2.292893219$$

Solution Steps

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

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

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}.

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

Square 2. Square \sqrt{2}.

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

Subtract 2 from 4 to get 2.

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

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

$$\frac{10-5\sqrt{2}+4\sqrt{2}-2\left(\sqrt{2}\right)^{2}}{2}$$

Combine -5\sqrt{2} and 4\sqrt{2} to get -\sqrt{2}.

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

The square of \sqrt{2} is 2.

$$\frac{10-\sqrt{2}-2\times 2}{2}$$

Multiply -2 and 2 to get -4.

$$\frac{10-\sqrt{2}-4}{2}$$

Subtract 4 from 10 to get 6.

$$\frac{6-\sqrt{2}}{2}$$