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

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Question

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

Evaluate

$\sqrt{5}-2\approx 0.236067977$

Solution Steps

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

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

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

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

Square $2$. Square $\sqrt{5}$.

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

Subtract $5$ from $4$ to get $-1$.

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

Anything divided by -1 gives its opposite. To find the opposite of $2-\sqrt{5}$, find the opposite of each term.

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

The opposite of $-\sqrt{5}$ is $\sqrt{5}$.

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

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