Solve frac{sqrt{5}+3}{sqrt{5}-2}=

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

Evaluate

$$5\sqrt{5}+11\approx 22.180339887$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{5}+3}{\sqrt{5}-2} by multiplying numerator and denominator by \sqrt{5}+2.

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

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}.

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

Square \sqrt{5}. Square 2.

$$\frac{\left(\sqrt{5}+3\right)\left(\sqrt{5}+2\right)}{5-4}$$

Subtract 4 from 5 to get 1.

$$\frac{\left(\sqrt{5}+3\right)\left(\sqrt{5}+2\right)}{1}$$

Anything divided by one gives itself.

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

Apply the distributive property by multiplying each term of \sqrt{5}+3 by each term of \sqrt{5}+2.

$$\left(\sqrt{5}\right)^{2}+2\sqrt{5}+3\sqrt{5}+6$$

The square of \sqrt{5} is 5.

$$5+2\sqrt{5}+3\sqrt{5}+6$$

Combine 2\sqrt{5} and 3\sqrt{5} to get 5\sqrt{5}.

$$5+5\sqrt{5}+6$$

Add 5 and 6 to get 11.

$$11+5\sqrt{5}$$