Rationalize the denominator of 2/(sqrt 7 + sqrt 2) and simplify the answer.

Step-by-step solution

1. 1

   Choose the conjugate, not just another copy of the denominator. With a sum of two radicals below, multiplying by $\sqrt7+\sqrt2$ again would produce a cross term $2\sqrt{14}$ and leave you worse off. The conjugate changes only the middle sign:

   $\text{conjugate of }\sqrt7+\sqrt2\ \text{is}\ \sqrt7-\sqrt2.$

2. 2

   Multiply numerator and denominator by that conjugate.

   $\frac{2}{\sqrt7+\sqrt2}\cdot\frac{\sqrt7-\sqrt2}{\sqrt7-\sqrt2}=\frac{2\left(\sqrt7-\sqrt2\right)}{\left(\sqrt7+\sqrt2\right)\left(\sqrt7-\sqrt2\right)}.$

3. 3

   Use the difference of squares below. $(a+b)(a-b)=a^2-b^2$, and squaring a square root removes it:

   $\left(\sqrt7\right)^{2}-\left(\sqrt2\right)^{2}=7-2=5.$

4. 4

   Write the simplified fraction.

   $\frac{2\left(\sqrt7-\sqrt2\right)}{5}=\frac{2\sqrt7-2\sqrt2}{5}.$

   The numerator's $2$ and the denominator's $5$ share no common factor, so nothing cancels — and cancelling the $2$ against a radical would be invalid anyway.

5. 5

   Check numerically. $\sqrt7=2.645751$, $\sqrt2=1.414214$, so the original is $2/4.059964=0.4926151$, and the rationalized form is $2(1.231538)/5=0.4926151$. They agree to every digit shown.