Question

Add $$\left(\dfrac{2}{3}\sqrt{7}-\dfrac{1}{2}\sqrt{2}+6\sqrt{11}\right)$$ and $$\left(\dfrac{1}{3}\sqrt{7}+\dfrac{3}{2}\sqrt{2}-\sqrt{11}\right)$$.

Answer 1

$$\left(\dfrac{2}{3}\sqrt{7}-\dfrac{1}{2}\sqrt{2}+6\sqrt{11}\right)$$ and $$\left(\dfrac{1}{3}\sqrt{7}+\dfrac{3}{2}\sqrt{2}-\sqrt{11}\right)$$

By adding both

$$=\left(\dfrac{2}{3}\sqrt{7}-\dfrac{1}{2}\sqrt{2}+6\sqrt{11}\right)+\left(\dfrac{1}{3}\sqrt{7}+\dfrac{3}{2}\sqrt{2}-\sqrt{11}\right)$$

On further calculation

$$=\left(\dfrac{2}{3}\sqrt{7}+\dfrac{1}{3}\sqrt{7}\right)+\left(-\dfrac{1}{2}\sqrt{2}+\dfrac{3}{2}\sqrt{2}\right)+\left(6\sqrt{11}-\sqrt{11}\right)$$

So we get

$$=\left(\dfrac{2}{3}+\dfrac{1}{3}\right)\sqrt{7}+\left(-\dfrac{1}{2}+\dfrac{3}{2}\right)\sqrt{2}+(6-1)\sqrt{11}$$.

By simplification

$$=\sqrt{7}+\sqrt{2}+5\sqrt{11}$$.