Solve frac{sqrt{10}+sqrt{15}}{sqrt{2}+sqrt{3}}

$$\frac { \sqrt { 10 } + \sqrt { 15 } } { \sqrt { 2 } + \sqrt { 3 } }$$

Evaluate

$$\sqrt{5}\approx 2.236067977$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{10}+\sqrt{15}}{\sqrt{2}+\sqrt{3}} by multiplying numerator and denominator by \sqrt{2}-\sqrt{3}.

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

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}.

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

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

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

Subtract 3 from 2 to get -1.

$$\frac{\left(\sqrt{10}+\sqrt{15}\right)\left(\sqrt{2}-\sqrt{3}\right)}{-1}$$

Anything divided by -1 gives its opposite.

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

Apply the distributive property by multiplying each term of \sqrt{10}+\sqrt{15} by each term of \sqrt{2}-\sqrt{3}.

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

Factor 10=2\times 5. Rewrite the square root of the product \sqrt{2\times 5} as the product of square roots \sqrt{2}\sqrt{5}.

$$-\left(\sqrt{2}\sqrt{5}\sqrt{2}-\sqrt{10}\sqrt{3}+\sqrt{15}\sqrt{2}-\sqrt{15}\sqrt{3}\right)$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$-\left(2\sqrt{5}-\sqrt{10}\sqrt{3}+\sqrt{15}\sqrt{2}-\sqrt{15}\sqrt{3}\right)$$

To multiply \sqrt{10} and \sqrt{3}, multiply the numbers under the square root.

$$-\left(2\sqrt{5}-\sqrt{30}+\sqrt{15}\sqrt{2}-\sqrt{15}\sqrt{3}\right)$$

To multiply \sqrt{15} and \sqrt{2}, multiply the numbers under the square root.

$$-\left(2\sqrt{5}-\sqrt{30}+\sqrt{30}-\sqrt{15}\sqrt{3}\right)$$

Combine -\sqrt{30} and \sqrt{30} to get 0.

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

Factor 15=3\times 5. Rewrite the square root of the product \sqrt{3\times 5} as the product of square roots \sqrt{3}\sqrt{5}.

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

Multiply \sqrt{3} and \sqrt{3} to get 3.

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

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

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

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

$$\sqrt{5}$$