Solve 1/1-sqrt{2}=

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

Evaluate

$$-\sqrt{2}-1\approx -2.414213562$$

Solution Steps

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

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

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}.

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

Square 1. Square \sqrt{2}.

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

Subtract 2 from 1 to get -1.

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

Anything divided by -1 gives its opposite. To find the opposite of 1+\sqrt{2}, find the opposite of each term.

$$-1-\sqrt{2}$$