Solve frac{4+sqrt{3}}{2+sqrt{5}}

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

Evaluate

$$\sqrt{15}+4\sqrt{5}-2\sqrt{3}-8\approx 1.353153641$$

Solution Steps

Rationalize the denominator of \frac{4+\sqrt{3}}{2+\sqrt{5}} by multiplying numerator and denominator by 2-\sqrt{5}.

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

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}.

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

Square 2. Square \sqrt{5}.

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

Subtract 5 from 4 to get -1.

$$\frac{\left(4+\sqrt{3}\right)\left(2-\sqrt{5}\right)}{-1}$$

Anything divided by -1 gives its opposite.

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

Apply the distributive property by multiplying each term of 4+\sqrt{3} by each term of 2-\sqrt{5}.

$$-\left(8-4\sqrt{5}+2\sqrt{3}-\sqrt{3}\sqrt{5}\right)$$

To multiply \sqrt{3} and \sqrt{5}, multiply the numbers under the square root.

$$-\left(8-4\sqrt{5}+2\sqrt{3}-\sqrt{15}\right)$$

To find the opposite of 8-4\sqrt{5}+2\sqrt{3}-\sqrt{15}, find the opposite of each term.

$$-8-\left(-4\sqrt{5}\right)-2\sqrt{3}-\left(-\sqrt{15}\right)$$

The opposite of -4\sqrt{5} is 4\sqrt{5}.

$$-8+4\sqrt{5}-2\sqrt{3}-\left(-\sqrt{15}\right)$$

The opposite of -\sqrt{15} is \sqrt{15}.

$$-8+4\sqrt{5}-2\sqrt{3}+\sqrt{15}$$