Quick sum
Is this surd sum exactly 3?
Age
16 to 18
Challenge level
Problem
According to my calculator, I see that
How could I prove that the answer is, indeed, exactly 3?
Did you know ... ?
Mathematicians often use calculating aids to help form an opinion concerning the likely numerical answer to a problem involving irrational numbers, but will always seek a full proof which does not rely on the calculator to complete the problem.
Mathematicians often use calculating aids to help form an opinion concerning the likely numerical answer to a problem involving irrational numbers, but will always seek a full proof which does not rely on the calculator to complete the problem.
Getting Started
Try rationalising the denominators.
Student Solutions
Let
When surds appear in the denominator the first step is almost always to rationalise the denominator. We see that
For each term the denominators multiply to $-1$. For example
Thus we have