Solve 2/sqrt{2-2}+frac{sqrt{2}+1}{sqrt{2}-1}-frac{sqrt{32}}{2}

$$\frac { 2 } { \sqrt { 2 } - 2 } + \frac { \sqrt { 2 } + 1 } { \sqrt { 2 } - 1 } - \frac { \sqrt { 32 } } { 2 }$$

Evaluate

$$1-\sqrt{2}\approx -0.414213562$$

Solution Steps

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

$$\frac{2\left(\sqrt{2}+2\right)}{\left(\sqrt{2}-2\right)\left(\sqrt{2}+2\right)}+\frac{\sqrt{2}+1}{\sqrt{2}-1}-\frac{\sqrt{32}}{2}$$

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}.

$$\frac{2\left(\sqrt{2}+2\right)}{\left(\sqrt{2}\right)^{2}-2^{2}}+\frac{\sqrt{2}+1}{\sqrt{2}-1}-\frac{\sqrt{32}}{2}$$

Square \sqrt{2}. Square 2.

$$\frac{2\left(\sqrt{2}+2\right)}{2-4}+\frac{\sqrt{2}+1}{\sqrt{2}-1}-\frac{\sqrt{32}}{2}$$

Subtract 4 from 2 to get -2.

$$\frac{2\left(\sqrt{2}+2\right)}{-2}+\frac{\sqrt{2}+1}{\sqrt{2}-1}-\frac{\sqrt{32}}{2}$$

Cancel out -2 and -2.

$$-\left(\sqrt{2}+2\right)+\frac{\sqrt{2}+1}{\sqrt{2}-1}-\frac{\sqrt{32}}{2}$$

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

$$-\left(\sqrt{2}+2\right)+\frac{\left(\sqrt{2}+1\right)\left(\sqrt{2}+1\right)}{\left(\sqrt{2}-1\right)\left(\sqrt{2}+1\right)}-\frac{\sqrt{32}}{2}$$

Consider \left(\sqrt{2}-1\right)\left(\sqrt{2}+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$-\left(\sqrt{2}+2\right)+\frac{\left(\sqrt{2}+1\right)\left(\sqrt{2}+1\right)}{\left(\sqrt{2}\right)^{2}-1^{2}}-\frac{\sqrt{32}}{2}$$

Square \sqrt{2}. Square 1.

$$-\left(\sqrt{2}+2\right)+\frac{\left(\sqrt{2}+1\right)\left(\sqrt{2}+1\right)}{2-1}-\frac{\sqrt{32}}{2}$$

Subtract 1 from 2 to get 1.

$$-\left(\sqrt{2}+2\right)+\frac{\left(\sqrt{2}+1\right)\left(\sqrt{2}+1\right)}{1}-\frac{\sqrt{32}}{2}$$

Anything divided by one gives itself.

$$-\left(\sqrt{2}+2\right)+\left(\sqrt{2}+1\right)\left(\sqrt{2}+1\right)-\frac{\sqrt{32}}{2}$$

Multiply \sqrt{2}+1 and \sqrt{2}+1 to get \left(\sqrt{2}+1\right)^{2}.

$$-\left(\sqrt{2}+2\right)+\left(\sqrt{2}+1\right)^{2}-\frac{\sqrt{32}}{2}$$

Factor 32=4^{2}\times 2. Rewrite the square root of the product \sqrt{4^{2}\times 2} as the product of square roots \sqrt{4^{2}}\sqrt{2}. Take the square root of 4^{2}.

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

Divide 4\sqrt{2} by 2 to get 2\sqrt{2}.

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

To find the opposite of \sqrt{2}+2, find the opposite of each term.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{2}+1\right)^{2}.

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

The square of \sqrt{2} is 2.

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

Add 2 and 1 to get 3.

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

Add -2 and 3 to get 1.

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

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

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

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

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

Factor

$$1-\sqrt{2}$$