Solve 3-frac{sqrt{2}}{(1-sqrt{5}}

$$3- \frac{ \sqrt{ 2 } }{ (1- \sqrt{ 5 } }$$

Evaluate

$$\frac{\sqrt{2}+\sqrt{10}+12}{4}\approx 4.144122806$$

Short Solution Steps

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

$$3-\frac{\sqrt{2}\left(1+\sqrt{5}\right)}{\left(1-\sqrt{5}\right)\left(1+\sqrt{5}\right)}$$

Consider \left(1-\sqrt{5}\right)\left(1+\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$3-\frac{\sqrt{2}\left(1+\sqrt{5}\right)}{1^{2}-\left(\sqrt{5}\right)^{2}}$$

Square 1. Square \sqrt{5}.

$$3-\frac{\sqrt{2}\left(1+\sqrt{5}\right)}{1-5}$$

Subtract 5 from 1 to get -4.

$$3-\frac{\sqrt{2}\left(1+\sqrt{5}\right)}{-4}$$

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

$$3-\frac{\sqrt{2}+\sqrt{2}\sqrt{5}}{-4}$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$3-\frac{\sqrt{2}+\sqrt{10}}{-4}$$

Multiply both numerator and denominator by -1.

$$3-\frac{-\sqrt{2}-\sqrt{10}}{4}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{4}{4}.

$$\frac{3\times 4}{4}-\frac{-\sqrt{2}-\sqrt{10}}{4}$$

Since \frac{3\times 4}{4} and \frac{-\sqrt{2}-\sqrt{10}}{4} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3\times 4-\left(-\sqrt{2}-\sqrt{10}\right)}{4}$$

Do the multiplications in 3\times 4-\left(-\sqrt{2}-\sqrt{10}\right).

$$\frac{12+\sqrt{2}+\sqrt{10}}{4}$$