Evaluate
\frac{5\left(\sqrt{5}-1\right)}{4}\approx 1.545084972
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\frac{5\left(\sqrt{5}-1\right)}{\left(\sqrt{5}+1\right)\left(\sqrt{5}-1\right)}
Rationalize the denominator of \frac{5}{\sqrt{5}+1} by multiplying numerator and denominator by \sqrt{5}-1.
\frac{5\left(\sqrt{5}-1\right)}{\left(\sqrt{5}\right)^{2}-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{5\left(\sqrt{5}-1\right)}{5-1}
Square \sqrt{5}. Square 1.
\frac{5\left(\sqrt{5}-1\right)}{4}
Subtract 1 from 5 to get 4.
\frac{5\sqrt{5}-5}{4}
Use the distributive property to multiply 5 by \sqrt{5}-1.
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