Solve \frac { 1 } { \sqrt { 3 } - 2 }

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Question

$$\frac { 1 } { \sqrt { 3 } - 2 }$$

Evaluate

$-\left(\sqrt{3}+2\right)\approx -3.732050808$

Solution Steps

Rationalize the denominator of $\frac{1}{\sqrt{3}-2}$ by multiplying numerator and denominator by $\sqrt{3}+2$.

$$\frac{\sqrt{3}+2}{\left(\sqrt{3}-2\right)\left(\sqrt{3}+2\right)}$$

Consider $\left(\sqrt{3}-2\right)\left(\sqrt{3}+2\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.

$$\frac{\sqrt{3}+2}{\left(\sqrt{3}\right)^{2}-2^{2}}$$

Square $\sqrt{3}$. Square $2$.

$$\frac{\sqrt{3}+2}{3-4}$$

Subtract $4$ from $3$ to get $-1$.

$$\frac{\sqrt{3}+2}{-1}$$

Anything divided by -1 gives its opposite. To find the opposite of $\sqrt{3}+2$, find the opposite of each term.

$$-\sqrt{3}-2$$

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