Solve frac{4+sqrt{5}}{2+sqrt{5}}

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

Evaluate

$$2\sqrt{5}-3\approx 1.472135955$$

Solution Steps

Rationalize the denominator of \frac{4+\sqrt{5}}{2+\sqrt{5}} by multiplying numerator and denominator by 2-\sqrt{5}.

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

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}.

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

Square 2. Square \sqrt{5}.

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

Subtract 5 from 4 to get -1.

$$\frac{\left(4+\sqrt{5}\right)\left(2-\sqrt{5}\right)}{-1}$$

Anything divided by -1 gives its opposite.

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

Apply the distributive property by multiplying each term of 4+\sqrt{5} by each term of 2-\sqrt{5}.

$$-\left(8-4\sqrt{5}+2\sqrt{5}-\left(\sqrt{5}\right)^{2}\right)$$

Combine -4\sqrt{5} and 2\sqrt{5} to get -2\sqrt{5}.

$$-\left(8-2\sqrt{5}-\left(\sqrt{5}\right)^{2}\right)$$

The square of \sqrt{5} is 5.

$$-\left(8-2\sqrt{5}-5\right)$$

Subtract 5 from 8 to get 3.

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

To find the opposite of 3-2\sqrt{5}, find the opposite of each term.

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

The opposite of -2\sqrt{5} is 2\sqrt{5}.

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