Solve 1divfrac{sqrt{5}-1}{2}

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

Evaluate

$$\frac{\sqrt{5}+1}{2}\approx 1.618033989$$

Solution Steps

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

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

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

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

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}.

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

Square \sqrt{5}. Square 1.

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

Subtract 1 from 5 to get 4.

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

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

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

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

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

Factor

$$\frac{\sqrt{5} + 1}{2} = 1.618033988749895$$