Solve 1/(2sqrt{5-1)}

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

Evaluate

$$\frac{2\sqrt{5}+1}{19}\approx 0.288007156$$

Solution Steps

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

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

Consider \left(2\sqrt{5}-1\right)\left(2\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}.

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

Expand \left(2\sqrt{5}\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{5} is 5.

$$\frac{2\sqrt{5}+1}{4\times 5-1^{2}}$$

Multiply 4 and 5 to get 20.

$$\frac{2\sqrt{5}+1}{20-1^{2}}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{2\sqrt{5}+1}{20-1}$$

Subtract 1 from 20 to get 19.

$$\frac{2\sqrt{5}+1}{19}$$