Solve 2*frac{1/2*frac{sqrt{2}}{2}}{frac{sqrt{3}+1}{2sqrt{2}}}

$$2 \times \frac{ \frac{ 1 }{ 2 } \times \frac{ \sqrt{ 2 } }{ 2 } }{ \frac{ \sqrt{ 3 } +1 }{ 2 \sqrt{ 2 } } }$$

Evaluate

$$\sqrt{3}-1\approx 0.732050808$$

Solution Steps

Multiply \frac{1}{2} times \frac{\sqrt{2}}{2} by multiplying numerator times numerator and denominator times denominator.

$$2\times \frac{\frac{\sqrt{2}}{2\times 2}}{\frac{\sqrt{3}+1}{2\sqrt{2}}}$$

Rationalize the denominator of \frac{\sqrt{3}+1}{2\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

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

Multiply 2 and 2 to get 4.

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

Divide \frac{\sqrt{2}}{2\times 2} by \frac{\left(\sqrt{3}+1\right)\sqrt{2}}{4} by multiplying \frac{\sqrt{2}}{2\times 2} by the reciprocal of \frac{\left(\sqrt{3}+1\right)\sqrt{2}}{4}.

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

Cancel out 2\times 2\sqrt{2} in both numerator and denominator.

$$2\times \frac{1}{\sqrt{3}+1}$$

Rationalize the denominator of \frac{1}{\sqrt{3}+1} by multiplying numerator and denominator by \sqrt{3}-1.

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

Consider \left(\sqrt{3}+1\right)\left(\sqrt{3}-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}.

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

Square \sqrt{3}. Square 1.

$$2\times \frac{\sqrt{3}-1}{3-1}$$

Subtract 1 from 3 to get 2.

$$2\times \frac{\sqrt{3}-1}{2}$$

Cancel out 2 and 2.

$$\sqrt{3}-1$$

Factor

$$\sqrt{3} - 1 = 0.732050808$$