Solve frac{sqrt{2}+sqrt{3}}{5+sqrt{6}}

$$\frac { \sqrt { 2 } + \sqrt { 3 } } { 5 + \sqrt { 6 } }$$

Evaluate

$$\frac{\left(5-\sqrt{6}\right)\left(\sqrt{2}+\sqrt{3}\right)}{19}\approx 0.422346292$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{2}+\sqrt{3}}{5+\sqrt{6}} by multiplying numerator and denominator by 5-\sqrt{6}.

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

Consider \left(5+\sqrt{6}\right)\left(5-\sqrt{6}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square 5. Square \sqrt{6}.

$$\frac{\left(\sqrt{2}+\sqrt{3}\right)\left(5-\sqrt{6}\right)}{25-6}$$

Subtract 6 from 25 to get 19.

$$\frac{\left(\sqrt{2}+\sqrt{3}\right)\left(5-\sqrt{6}\right)}{19}$$

Apply the distributive property by multiplying each term of \sqrt{2}+\sqrt{3} by each term of 5-\sqrt{6}.

$$\frac{5\sqrt{2}-\sqrt{2}\sqrt{6}+5\sqrt{3}-\sqrt{3}\sqrt{6}}{19}$$

Factor 6=2\times 3. Rewrite the square root of the product \sqrt{2\times 3} as the product of square roots \sqrt{2}\sqrt{3}.

$$\frac{5\sqrt{2}-\sqrt{2}\sqrt{2}\sqrt{3}+5\sqrt{3}-\sqrt{3}\sqrt{6}}{19}$$

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

$$\frac{5\sqrt{2}-2\sqrt{3}+5\sqrt{3}-\sqrt{3}\sqrt{6}}{19}$$

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

$$\frac{5\sqrt{2}+3\sqrt{3}-\sqrt{3}\sqrt{6}}{19}$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

$$\frac{5\sqrt{2}+3\sqrt{3}-\sqrt{3}\sqrt{3}\sqrt{2}}{19}$$

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

$$\frac{5\sqrt{2}+3\sqrt{3}-3\sqrt{2}}{19}$$

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

$$\frac{2\sqrt{2}+3\sqrt{3}}{19}$$