Solution of the Week #435 - A Satisfying Cancellation

The complicated expression turns out to have the value of precisely 3.

Call the sum of the two square roots ‘S’, and the difference ‘D’, the expression becomes S*sqrt(D/S)

Although S*D doesn’t appear anywhere in the expression, we will evaluate it in case it becomes useful later.

(sqrt(11)+sqrt(2))* (sqrt(11)-sqrt(2)) = 11+sqrt(22)–sqrt(22)-2 = 9.

If we look at the fraction under the square root, we can multiply top and bottom by D without changing its value. The top becomes D^2 and the bottom becomes 9. Then the square root (D^2/9) means that the whole square root expression can be replaced with D/3. Then the entire expression is just S*D/3, which of course is just 9/3 = 3.

Alternatively, change the left hand term S into the square root of S^2. Then you can combine the two square roots so that the overall expression is sqrt(S^2*D/S) or sqrt(S*D). As we saw above S*D is just 9, so sqrt(S*D)=3.