Solve {l}{sqrt{5}-3}{+1/sqrt{5-2}}

$$\left. \begin{array} { l } { \sqrt { 5 } - 3 } \\ { + \frac { 1 } { \sqrt { 5 } - 2 } } \end{array} \right.$$

Evaluate

$$2\sqrt{5}-1\approx 3.472135955$$

Solution Steps

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

$$\sqrt{5}-3+\frac{\sqrt{5}+2}{\left(\sqrt{5}-2\right)\left(\sqrt{5}+2\right)}$$

Consider \left(\sqrt{5}-2\right)\left(\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}.

$$\sqrt{5}-3+\frac{\sqrt{5}+2}{\left(\sqrt{5}\right)^{2}-2^{2}}$$

Square \sqrt{5}. Square 2.

$$\sqrt{5}-3+\frac{\sqrt{5}+2}{5-4}$$

Subtract 4 from 5 to get 1.

$$\sqrt{5}-3+\frac{\sqrt{5}+2}{1}$$

Anything divided by one gives itself.

$$\sqrt{5}-3+\sqrt{5}+2$$

Combine \sqrt{5} and \sqrt{5} to get 2\sqrt{5}.

$$2\sqrt{5}-3+2$$

Add -3 and 2 to get -1.

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