Solve frac{sqrt{12}}{sqrt{11}-3}+frac{sqrt{3}}{sqrt{11}+3}

$$\frac{ \sqrt{ 12 } }{ \sqrt{ 11 } -3 } + \frac{ \sqrt{ 3 } }{ \sqrt{ 11 } +3 }$$

Evaluate

$$\frac{3\left(\sqrt{3}+\sqrt{33}\right)}{2}\approx 11.214920181$$

Short Solution Steps

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$\frac{2\sqrt{3}}{\sqrt{11}-3}+\frac{\sqrt{3}}{\sqrt{11}+3}$$

Rationalize the denominator of \frac{2\sqrt{3}}{\sqrt{11}-3} by multiplying numerator and denominator by \sqrt{11}+3.

$$\frac{2\sqrt{3}\left(\sqrt{11}+3\right)}{\left(\sqrt{11}-3\right)\left(\sqrt{11}+3\right)}+\frac{\sqrt{3}}{\sqrt{11}+3}$$

Consider \left(\sqrt{11}-3\right)\left(\sqrt{11}+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}.

$$\frac{2\sqrt{3}\left(\sqrt{11}+3\right)}{\left(\sqrt{11}\right)^{2}-3^{2}}+\frac{\sqrt{3}}{\sqrt{11}+3}$$

Square \sqrt{11}. Square 3.

$$\frac{2\sqrt{3}\left(\sqrt{11}+3\right)}{11-9}+\frac{\sqrt{3}}{\sqrt{11}+3}$$

Subtract 9 from 11 to get 2.

$$\frac{2\sqrt{3}\left(\sqrt{11}+3\right)}{2}+\frac{\sqrt{3}}{\sqrt{11}+3}$$

Rationalize the denominator of \frac{\sqrt{3}}{\sqrt{11}+3} by multiplying numerator and denominator by \sqrt{11}-3.

$$\frac{2\sqrt{3}\left(\sqrt{11}+3\right)}{2}+\frac{\sqrt{3}\left(\sqrt{11}-3\right)}{\left(\sqrt{11}+3\right)\left(\sqrt{11}-3\right)}$$

Consider \left(\sqrt{11}+3\right)\left(\sqrt{11}-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}.

$$\frac{2\sqrt{3}\left(\sqrt{11}+3\right)}{2}+\frac{\sqrt{3}\left(\sqrt{11}-3\right)}{\left(\sqrt{11}\right)^{2}-3^{2}}$$

Square \sqrt{11}. Square 3.

$$\frac{2\sqrt{3}\left(\sqrt{11}+3\right)}{2}+\frac{\sqrt{3}\left(\sqrt{11}-3\right)}{11-9}$$

Subtract 9 from 11 to get 2.

$$\frac{2\sqrt{3}\left(\sqrt{11}+3\right)}{2}+\frac{\sqrt{3}\left(\sqrt{11}-3\right)}{2}$$

Since \frac{2\sqrt{3}\left(\sqrt{11}+3\right)}{2} and \frac{\sqrt{3}\left(\sqrt{11}-3\right)}{2} have the same denominator, add them by adding their numerators.

$$\frac{2\sqrt{3}\left(\sqrt{11}+3\right)+\sqrt{3}\left(\sqrt{11}-3\right)}{2}$$

Do the multiplications in 2\sqrt{3}\left(\sqrt{11}+3\right)+\sqrt{3}\left(\sqrt{11}-3\right).

$$\frac{2\sqrt{33}+6\sqrt{3}+\sqrt{33}-3\sqrt{3}}{2}$$

Do the calculations in 2\sqrt{33}+6\sqrt{3}+\sqrt{33}-3\sqrt{3}.

$$\frac{3\sqrt{33}+3\sqrt{3}}{2}$$