Solve 1/sqrt{5-1/2}+frac{sqrt{5}-1}{2}

$$\frac { 1 } { \frac { \sqrt { 5 } - 1 } { 2 } } + \frac { \sqrt { 5 } - 1 } { 2 }$$

Evaluate

$$\sqrt{5}\approx 2.236067977$$

Short Solution Steps

Divide 1 by \frac{\sqrt{5}-1}{2} by multiplying 1 by the reciprocal of \frac{\sqrt{5}-1}{2}.

$$\frac{2}{\sqrt{5}-1}+\frac{\sqrt{5}-1}{2}$$

Rationalize the denominator of \frac{2}{\sqrt{5}-1} by multiplying numerator and denominator by \sqrt{5}+1.

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

Consider \left(\sqrt{5}-1\right)\left(\sqrt{5}+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}.

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

Square \sqrt{5}. Square 1.

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

Subtract 1 from 5 to get 4.

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

Divide 2\left(\sqrt{5}+1\right) by 4 to get \frac{1}{2}\left(\sqrt{5}+1\right).

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

Use the distributive property to multiply \frac{1}{2} by \sqrt{5}+1.

$$\frac{1}{2}\sqrt{5}+\frac{1}{2}+\frac{\sqrt{5}-1}{2}$$

Since \frac{1}{2} and \frac{\sqrt{5}-1}{2} have the same denominator, add them by adding their numerators.

$$\frac{1}{2}\sqrt{5}+\frac{1+\sqrt{5}-1}{2}$$

Do the calculations in 1+\sqrt{5}-1.

$$\frac{1}{2}\sqrt{5}+\frac{\sqrt{5}}{2}$$

Combine \frac{1}{2}\sqrt{5} and \frac{\sqrt{5}}{2} to get \sqrt{5}.

$$\sqrt{5}$$