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

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

Evaluate

$$-\left(\sqrt{6}+\sqrt{10}+\sqrt{15}+2\right)\approx -11.484750749$$

Solution Steps

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

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

Consider \left(\sqrt{2}-\sqrt{3}\right)\left(\sqrt{2}+\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square \sqrt{2}. Square \sqrt{3}.

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

Subtract 3 from 2 to get -1.

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

Anything divided by -1 gives its opposite.

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

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

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

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$-\left(\sqrt{10}+\sqrt{5}\sqrt{3}+\left(\sqrt{2}\right)^{2}+\sqrt{2}\sqrt{3}\right)$$

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

$$-\left(\sqrt{10}+\sqrt{15}+\left(\sqrt{2}\right)^{2}+\sqrt{2}\sqrt{3}\right)$$

The square of \sqrt{2} is 2.

$$-\left(\sqrt{10}+\sqrt{15}+2+\sqrt{2}\sqrt{3}\right)$$

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$-\left(\sqrt{10}+\sqrt{15}+2+\sqrt{6}\right)$$

To find the opposite of \sqrt{10}+\sqrt{15}+2+\sqrt{6}, find the opposite of each term.

$$-\sqrt{10}-\sqrt{15}-2-\sqrt{6}$$