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

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

Evaluate

$$\frac{\sqrt{5}+1}{4}\approx 0.809016994$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{5} is 5.

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

Multiply 4 and 5 to get 20.

$$\frac{2\left(2\sqrt{5}+2\right)}{20-2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{2\left(2\sqrt{5}+2\right)}{20-4}$$

Subtract 4 from 20 to get 16.

$$\frac{2\left(2\sqrt{5}+2\right)}{16}$$

Divide 2\left(2\sqrt{5}+2\right) by 16 to get \frac{1}{8}\left(2\sqrt{5}+2\right).

$$\frac{1}{8}\left(2\sqrt{5}+2\right)$$

Use the distributive property to multiply \frac{1}{8} by 2\sqrt{5}+2.

$$\frac{1}{8}\times 2\sqrt{5}+\frac{1}{8}\times 2$$

Multiply \frac{1}{8} and 2 to get \frac{2}{8}.

$$\frac{2}{8}\sqrt{5}+\frac{1}{8}\times 2$$

Reduce the fraction \frac{2}{8} to lowest terms by extracting and canceling out 2.

$$\frac{1}{4}\sqrt{5}+\frac{1}{8}\times 2$$

Multiply \frac{1}{8} and 2 to get \frac{2}{8}.

$$\frac{1}{4}\sqrt{5}+\frac{2}{8}$$

Reduce the fraction \frac{2}{8} to lowest terms by extracting and canceling out 2.

$$\frac{1}{4}\sqrt{5}+\frac{1}{4}$$