Solve frac{1/sqrt{2}}{1-1/sqrt{2}}

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

Evaluate

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

Short Solution Steps

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

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

The square of \sqrt{2} is 2.

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

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

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

The square of \sqrt{2} is 2.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2}{2}.

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

Since \frac{2}{2} and \frac{\sqrt{2}}{2} have the same denominator, subtract them by subtracting their numerators.

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

Divide \frac{\sqrt{2}}{2} by \frac{2-\sqrt{2}}{2} by multiplying \frac{\sqrt{2}}{2} by the reciprocal of \frac{2-\sqrt{2}}{2}.

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

Cancel out 2 in both numerator and denominator.

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

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

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

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

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

Expand \left(-\sqrt{2}\right)^{2}.

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

Calculate -1 to the power of 2 and get 1.

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

The square of \sqrt{2} is 2.

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

Multiply 1 and 2 to get 2.

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

Calculate 2 to the power of 2 and get 4.

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

Subtract 4 from 2 to get -2.

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

Use the distributive property to multiply \sqrt{2} by -\sqrt{2}-2.

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

The square of \sqrt{2} is 2.

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

Divide each term of -2-2\sqrt{2} by -2 to get 1+\sqrt{2}.

$$1+\sqrt{2}$$