Solve frac{2-sqrt{3}}{4-sqrt{6}}

$$\frac { 2 - \sqrt { 3 } } { 4 - \sqrt { 6 } }$$

Evaluate

$$\frac{\sqrt{6}}{5}-\frac{2\sqrt{3}}{5}-\frac{3\sqrt{2}}{10}+\frac{4}{5}\approx 0.172813557$$

Solution Steps

Rationalize the denominator of \frac{2-\sqrt{3}}{4-\sqrt{6}} by multiplying numerator and denominator by 4+\sqrt{6}.

$$\frac{\left(2-\sqrt{3}\right)\left(4+\sqrt{6}\right)}{\left(4-\sqrt{6}\right)\left(4+\sqrt{6}\right)}$$

Consider \left(4-\sqrt{6}\right)\left(4+\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}.

$$\frac{\left(2-\sqrt{3}\right)\left(4+\sqrt{6}\right)}{4^{2}-\left(\sqrt{6}\right)^{2}}$$

Square 4. Square \sqrt{6}.

$$\frac{\left(2-\sqrt{3}\right)\left(4+\sqrt{6}\right)}{16-6}$$

Subtract 6 from 16 to get 10.

$$\frac{\left(2-\sqrt{3}\right)\left(4+\sqrt{6}\right)}{10}$$

Apply the distributive property by multiplying each term of 2-\sqrt{3} by each term of 4+\sqrt{6}.

$$\frac{8+2\sqrt{6}-4\sqrt{3}-\sqrt{3}\sqrt{6}}{10}$$

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}.

$$\frac{8+2\sqrt{6}-4\sqrt{3}-\sqrt{3}\sqrt{3}\sqrt{2}}{10}$$

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

$$\frac{8+2\sqrt{6}-4\sqrt{3}-3\sqrt{2}}{10}$$