Surds based problem

Question

Find the rationalizing factor of $\sqrt{2} +\sqrt7 -\sqrt{10}$. How to do the sum?

Answer 1

Use the following statement.$$2(a^2b^2+a^2c^2+b^2c^2)-a^4-b^4-c^4=(a+b+c)(a+b-c)(a+c-b)(b+c-a)$$

Answer 2

$$\begin{array}{rcl} \sqrt2+\sqrt7-\sqrt{10} &=& \dfrac{\sqrt2+\sqrt7-\sqrt{10}}1 \\ &=& \dfrac{(\sqrt2+\sqrt7)^2-10}{\sqrt2+\sqrt7+\sqrt{10}} \\ &=& \dfrac{43+2\sqrt{14}}{\sqrt2+\sqrt7+\sqrt{10}} \\ &=& \dfrac{1793}{(\sqrt2+\sqrt7+\sqrt{10})(43-2\sqrt{14})} \\ \end{array}$$

Thus $(\sqrt2+\sqrt7+\sqrt{10})(43-2\sqrt{14})$ is what you seek.