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

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

Evaluate

$$\frac{3\sqrt{2}}{2}\approx 2.121320344$$

Short Solution Steps

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

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

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}.

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

Square 2. Square \sqrt{2}.

$$\left(2+\sqrt{2}\right)\times \frac{2+\sqrt{2}}{4-2}-\frac{1}{2+\sqrt{2}}-1\left(2+\sqrt{2}\right)$$

Subtract 2 from 4 to get 2.

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

Express \left(2+\sqrt{2}\right)\times \frac{2+\sqrt{2}}{2} as a single fraction.

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

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

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

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}.

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

Square 2. Square \sqrt{2}.

$$\frac{\left(2+\sqrt{2}\right)\left(2+\sqrt{2}\right)}{2}-\frac{2-\sqrt{2}}{4-2}-1\left(2+\sqrt{2}\right)$$

Subtract 2 from 4 to get 2.

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

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

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

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

$$\frac{4+2\sqrt{2}+2\sqrt{2}+2-2+\sqrt{2}}{2}-1\left(2+\sqrt{2}\right)$$

Do the calculations in 4+2\sqrt{2}+2\sqrt{2}+2-2+\sqrt{2}.

$$\frac{4+5\sqrt{2}}{2}-1\left(2+\sqrt{2}\right)$$

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

$$\frac{4+5\sqrt{2}}{2}-\frac{2\left(2+\sqrt{2}\right)}{2}$$

Since \frac{4+5\sqrt{2}}{2} and \frac{2\left(2+\sqrt{2}\right)}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4+5\sqrt{2}-2\left(2+\sqrt{2}\right)}{2}$$

Do the multiplications in 4+5\sqrt{2}-2\left(2+\sqrt{2}\right).

$$\frac{4+5\sqrt{2}-4-2\sqrt{2}}{2}$$

Do the calculations in 4+5\sqrt{2}-4-2\sqrt{2}.

$$\frac{3\sqrt{2}}{2}$$