Write `sqrt(-97)` in the form `bi` where `b` is a real number.

Expert Answers
degeneratecircle eNotes educator| Certified Educator

We use the fact that `sqrt(ab)=sqrt(a)sqrt(b),` which is valid for the numbers `a=-1,b=97.`

We get `sqrt(-97)=sqrt((-1)(97))=sqrt(-1)sqrt(97)=isqrt(97).`

This can't be simplified any further, so the answer is `isqrt(97),` and in particular, our value of `b` is `sqrt(97).`

A little aside: It doesn't matter in this particular proof, but it's worth keeping in mind that the relation `sqrt(ab)=sqrt(a)sqrt(b)` is not always valid, depending on what `a` and `b` are. There are a lot of false "proofs" of things like `1=-1` that use that relationship when it doesn't apply, and then get nonsense from it.