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

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

Evaluate

$$\frac{3\left(\sqrt{2}+4\right)}{7}\approx 2.320377241$$

Short Solution Steps

Since \frac{2-\sqrt{2}}{2} and \frac{\sqrt{2}+1}{2} have the same denominator, add them by adding their numerators.

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

Do the calculations in 2-\sqrt{2}+\sqrt{2}+1.

$$\frac{\frac{3}{2}}{1-\frac{\frac{2-\sqrt{2}}{2}\left(\sqrt{2}+1\right)}{2}}$$

Express \frac{2-\sqrt{2}}{2}\left(\sqrt{2}+1\right) as a single fraction.

$$\frac{\frac{3}{2}}{1-\frac{\frac{\left(2-\sqrt{2}\right)\left(\sqrt{2}+1\right)}{2}}{2}}$$

Express \frac{\frac{\left(2-\sqrt{2}\right)\left(\sqrt{2}+1\right)}{2}}{2} as a single fraction.

$$\frac{\frac{3}{2}}{1-\frac{\left(2-\sqrt{2}\right)\left(\sqrt{2}+1\right)}{2\times 2}}$$

Multiply 2 and 2 to get 4.

$$\frac{\frac{3}{2}}{1-\frac{\left(2-\sqrt{2}\right)\left(\sqrt{2}+1\right)}{4}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{4}{4}.

$$\frac{\frac{3}{2}}{\frac{4}{4}-\frac{\left(2-\sqrt{2}\right)\left(\sqrt{2}+1\right)}{4}}$$

Since \frac{4}{4} and \frac{\left(2-\sqrt{2}\right)\left(\sqrt{2}+1\right)}{4} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{3}{2}}{\frac{4-\left(2-\sqrt{2}\right)\left(\sqrt{2}+1\right)}{4}}$$

Do the multiplications in 4-\left(2-\sqrt{2}\right)\left(\sqrt{2}+1\right).

$$\frac{\frac{3}{2}}{\frac{4-2\sqrt{2}-2+2+\sqrt{2}}{4}}$$

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

$$\frac{\frac{3}{2}}{\frac{4-\sqrt{2}}{4}}$$

Divide \frac{3}{2} by \frac{4-\sqrt{2}}{4} by multiplying \frac{3}{2} by the reciprocal of \frac{4-\sqrt{2}}{4}.

$$\frac{3\times 4}{2\left(4-\sqrt{2}\right)}$$

Cancel out 2 in both numerator and denominator.

$$\frac{2\times 3}{-\sqrt{2}+4}$$

Rationalize the denominator of \frac{2\times 3}{-\sqrt{2}+4} by multiplying numerator and denominator by -\sqrt{2}-4.

$$\frac{2\times 3\left(-\sqrt{2}-4\right)}{\left(-\sqrt{2}+4\right)\left(-\sqrt{2}-4\right)}$$

Consider \left(-\sqrt{2}+4\right)\left(-\sqrt{2}-4\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{2\times 3\left(-\sqrt{2}-4\right)}{\left(-\sqrt{2}\right)^{2}-4^{2}}$$

Multiply 2 and 3 to get 6.

$$\frac{6\left(-\sqrt{2}-4\right)}{\left(-\sqrt{2}\right)^{2}-4^{2}}$$

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

$$\frac{6\left(-\sqrt{2}-4\right)}{\left(-1\right)^{2}\left(\sqrt{2}\right)^{2}-4^{2}}$$

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

$$\frac{6\left(-\sqrt{2}-4\right)}{1\left(\sqrt{2}\right)^{2}-4^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{6\left(-\sqrt{2}-4\right)}{1\times 2-4^{2}}$$

Multiply 1 and 2 to get 2.

$$\frac{6\left(-\sqrt{2}-4\right)}{2-4^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{6\left(-\sqrt{2}-4\right)}{2-16}$$

Subtract 16 from 2 to get -14.

$$\frac{6\left(-\sqrt{2}-4\right)}{-14}$$

Divide 6\left(-\sqrt{2}-4\right) by -14 to get -\frac{3}{7}\left(-\sqrt{2}-4\right).

$$-\frac{3}{7}\left(-\sqrt{2}-4\right)$$

Use the distributive property to multiply -\frac{3}{7} by -\sqrt{2}-4.

$$-\frac{3}{7}\left(-1\right)\sqrt{2}-\frac{3}{7}\left(-4\right)$$

Multiply -\frac{3}{7} and -1 to get \frac{3}{7}.

$$\frac{3}{7}\sqrt{2}-\frac{3}{7}\left(-4\right)$$

Express -\frac{3}{7}\left(-4\right) as a single fraction.

$$\frac{3}{7}\sqrt{2}+\frac{-3\left(-4\right)}{7}$$

Multiply -3 and -4 to get 12.

$$\frac{3}{7}\sqrt{2}+\frac{12}{7}$$

Factor

$$\frac{3 {(\sqrt{2} + 4)}}{7} = 2.3203772410170407$$