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

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

Evaluate

$$3\sqrt{2}-4\approx 0.242640687$$

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 -3\sqrt{2}.

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

The square of \sqrt{2} is 2.

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

Add 2 and 2 to get 4.

$$-\left(4-3\sqrt{2}\right)$$

To find the opposite of 4-3\sqrt{2}, find the opposite of each term.

$$-4-\left(-3\sqrt{2}\right)$$

The opposite of -3\sqrt{2} is 3\sqrt{2}.

$$-4+3\sqrt{2}$$

Factor

$$3 \sqrt{2} - 4 = 0.242640687$$