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

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

Evaluate

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Short Solution Steps

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

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

The square of \sqrt{2} is 2.

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

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

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

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}.

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

Square 1. Square \sqrt{2}.

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

Subtract 2 from 1 to get -1.

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

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

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

To find the opposite of -1-\left(-\sqrt{2}\right), find the opposite of each term.

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

The opposite of -1 is 1.

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

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

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

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

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

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}.

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

Square 2. Square \sqrt{2}.

$$\frac{\sqrt{2}}{2}+1-\sqrt{2}-\frac{2-\sqrt{2}}{4-2}$$

Subtract 2 from 4 to get 2.

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

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

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

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

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

Combine \frac{\sqrt{2}}{2} and -\sqrt{2} to get -\frac{1}{2}\sqrt{2}.

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

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

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

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

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

Do the multiplications in 2-\left(2-\sqrt{2}\right).

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

Do the calculations in 2-2+\sqrt{2}.

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

Combine -\frac{1}{2}\sqrt{2} and \frac{\sqrt{2}}{2} to get 0.

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Factor

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