Solve frac{2+sqrt{2}}{1+sqrt{2}}

$$\frac { 2 + \sqrt { 2 } } { 1 + \sqrt { 2 } }$$

Evaluate

$$\sqrt{2}\approx 1.414213562$$

Solution Steps

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

$$\frac{\left(2+\sqrt{2}\right)\left(1-\sqrt{2}\right)}{\left(1+\sqrt{2}\right)\left(1-\sqrt{2}\right)}$$

Consider \left(1+\sqrt{2}\right)\left(1-\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}.

$$\frac{\left(2+\sqrt{2}\right)\left(1-\sqrt{2}\right)}{1^{2}-\left(\sqrt{2}\right)^{2}}$$

Square 1. Square \sqrt{2}.

$$\frac{\left(2+\sqrt{2}\right)\left(1-\sqrt{2}\right)}{1-2}$$

Subtract 2 from 1 to get -1.

$$\frac{\left(2+\sqrt{2}\right)\left(1-\sqrt{2}\right)}{-1}$$

Anything divided by -1 gives its opposite.

$$-\left(2+\sqrt{2}\right)\left(1-\sqrt{2}\right)$$

Apply the distributive property by multiplying each term of 2+\sqrt{2} by each term of 1-\sqrt{2}.

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

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

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

The square of \sqrt{2} is 2.

$$-\left(2-\sqrt{2}-2\right)$$

Subtract 2 from 2 to get 0.

$$-\left(-\sqrt{2}\right)$$

The opposite of -\sqrt{2} is \sqrt{2}.

$$\sqrt{2}$$