sqrt(18) - sqrt(17)

Question

How does this expression equal

1/(sqrt(18) + sqrt(17)) ?
How would you change it like that?
And why is this good to use when not using a calculator?

Answer 1

(a-b)(a+b) = a^2-b^2, so

(√18-√17) * (√18+√17)/(√18+√17) = (18-17)/√(18+√17)

A calculator has roundoff error. √18 is very close to √17. Not too close in this case, but √188888 - √188887 are very close, and (floating-point) precision will be lost when trying to include the result with other numbers not so small.

If you add them, the roundoff occurs farther out, making it less important.