In our last installment, we found that could be simplified using our formula in one step, only if is a perfect square and b is not. So now we must ask if we can simplify the expression in question when this condition is not satisfied in two or three or any finite number of applications of our formula? Let’s try a specific example first. .
In order to apply our formula to this result, we rewrite the result as .
Applying the formula to the first term, we get
. Similarly, for the second term, we get , and the sum of these two results is , so we are back where we started.
Is this circularity independent of the values of a and b ?
If we rewrite the right side of the formula so that each of the two fractions is the sum of a rational and a radical, they will have the same format as the radicand of the left side of the formula, so we can apply the original formula to the result. If we do exactly what we did with the specific example, we get . Applying our formula to the first term, we get
. Similarly, the second term, and the sum of these two results is , so we are back where we started. The same is true for . Therefore, two successive applications of this formula will return an expression of the form back to the original expression.
Hence, any expression of this form that cannot be simplified with one application of the formula cannot be simplified at all by using this formula. And no other formula will work either, as any such formula is an identity derived by algebraic manipulation, as this one is, and is therefore equivalent to this one.