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

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

Evaluate

$$\sqrt{2}\left(\sqrt{3}-1\right)\approx 1.03527618$$

Solution Steps

Divide \sqrt{2} by \frac{\sqrt{3}+1}{2} by multiplying \sqrt{2} by the reciprocal of \frac{\sqrt{3}+1}{2}.

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

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

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

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}.

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

Square \sqrt{3}. Square 1.

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

Subtract 1 from 3 to get 2.

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

Use the distributive property to multiply \sqrt{2}\times 2 by \sqrt{3}-1.

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

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{2\sqrt{6}-\sqrt{2}\times 2}{2}$$

Multiply -1 and 2 to get -2.

$$\frac{2\sqrt{6}-2\sqrt{2}}{2}$$

Divide each term of 2\sqrt{6}-2\sqrt{2} by 2 to get \sqrt{6}-\sqrt{2}.

$$\sqrt{6}-\sqrt{2}$$