Rationalize the denominator and simplify the expression 2 divided by the quantity square root of 7 plus square root of 2.

Step-by-step solution

1. 1

   See why a single multiplication will not work. Multiplying top and bottom by $\sqrt{7}$ alone leaves $\sqrt{14}$ behind in the denominator. A sum of two different radicals needs the conjugate, because only the difference-of-squares pattern kills both roots at once.

2. 2

   Multiply by the conjugate over itself. The conjugate of $\sqrt{7}+\sqrt{2}$ is $\sqrt{7}-\sqrt{2}$. Multiplying by $\dfrac{\sqrt7-\sqrt2}{\sqrt7-\sqrt2}$ is multiplying by $1$, so the value never changes:

   $\frac{2}{\sqrt{7}+\sqrt{2}}\cdot\frac{\sqrt{7}-\sqrt{2}}{\sqrt{7}-\sqrt{2}}$

3. 3

   Expand the denominator with $(a+b)(a-b)=a^2-b^2$. The cross terms cancel, which is exactly the point:

   $(\sqrt{7}+\sqrt{2})(\sqrt{7}-\sqrt{2}) = (\sqrt7)^2-(\sqrt2)^2 = 7-2 = 5$

4. 4

   Distribute in the numerator.

   $2(\sqrt{7}-\sqrt{2}) = 2\sqrt{7}-2\sqrt{2}$

   The result is $\dfrac{2\sqrt7-2\sqrt2}{5}$.

5. 5

   Check whether it reduces further. The greatest common factor of $2$, $2$ and $5$ is $1$, so nothing cancels; you may also write it as $\dfrac{2(\sqrt7-\sqrt2)}{5}$.

6. 6

   Verify numerically. $\sqrt7\approx2.6458$ and $\sqrt2\approx1.4142$, so the original is $\tfrac{2}{4.0600}\approx0.4926$, and the answer is $\tfrac{2(1.2316)}{5}\approx0.4926$. They agree.