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

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

Evaluate

$$-\left(\sqrt{2}-2\sqrt{3}\right)\approx 2.049888053$$

Solution Steps

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

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

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}.

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

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

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

Subtract 3 from 2 to get -1.

$$\frac{\left(4+\sqrt{6}\right)\left(\sqrt{2}-\sqrt{3}\right)}{-1}$$

Anything divided by -1 gives its opposite.

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

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

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

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}.

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

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

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

Combine -4\sqrt{3} and 2\sqrt{3} to get -2\sqrt{3}.

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

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}.

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

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

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

Combine 4\sqrt{2} and -3\sqrt{2} to get \sqrt{2}.

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

To find the opposite of \sqrt{2}-2\sqrt{3}, find the opposite of each term.

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

The opposite of -2\sqrt{3} is 2\sqrt{3}.

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