Solve frac{sqrt{3}}{sqrt{6}+sqrt{7}}

$$\frac { \sqrt { 3 } } { \sqrt { 6 } + \sqrt { 7 } }$$

Evaluate

$$\sqrt{21}-3\sqrt{2}\approx 0.339935008$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{3}}{\sqrt{6}+\sqrt{7}} by multiplying numerator and denominator by \sqrt{6}-\sqrt{7}.

$$\frac{\sqrt{3}\left(\sqrt{6}-\sqrt{7}\right)}{\left(\sqrt{6}+\sqrt{7}\right)\left(\sqrt{6}-\sqrt{7}\right)}$$

Consider \left(\sqrt{6}+\sqrt{7}\right)\left(\sqrt{6}-\sqrt{7}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\sqrt{3}\left(\sqrt{6}-\sqrt{7}\right)}{\left(\sqrt{6}\right)^{2}-\left(\sqrt{7}\right)^{2}}$$

Square \sqrt{6}. Square \sqrt{7}.

$$\frac{\sqrt{3}\left(\sqrt{6}-\sqrt{7}\right)}{6-7}$$

Subtract 7 from 6 to get -1.

$$\frac{\sqrt{3}\left(\sqrt{6}-\sqrt{7}\right)}{-1}$$

Anything divided by -1 gives its opposite.

$$-\sqrt{3}\left(\sqrt{6}-\sqrt{7}\right)$$

Use the distributive property to multiply \sqrt{3} by \sqrt{6}-\sqrt{7}.

$$-\left(\sqrt{3}\sqrt{6}-\sqrt{3}\sqrt{7}\right)$$

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}.

$$-\left(\sqrt{3}\sqrt{3}\sqrt{2}-\sqrt{3}\sqrt{7}\right)$$

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

$$-\left(3\sqrt{2}-\sqrt{3}\sqrt{7}\right)$$

To multiply \sqrt{3} and \sqrt{7}, multiply the numbers under the square root.

$$-\left(3\sqrt{2}-\sqrt{21}\right)$$

To find the opposite of 3\sqrt{2}-\sqrt{21}, find the opposite of each term.

$$-3\sqrt{2}-\left(-\sqrt{21}\right)$$

The opposite of -\sqrt{21} is \sqrt{21}.

$$-3\sqrt{2}+\sqrt{21}$$