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

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

Evaluate

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

Solution Steps

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

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

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}.

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

Square 2. Square \sqrt{5}.

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

Subtract 5 from 4 to get -1.

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

Anything divided by -1 gives its opposite.

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

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

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

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

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

The square of \sqrt{5} is 5.

$$-\left(6+5\sqrt{5}+5\right)$$

Add 6 and 5 to get 11.

$$-\left(11+5\sqrt{5}\right)$$

To find the opposite of 11+5\sqrt{5}, find the opposite of each term.

$$-11-5\sqrt{5}$$