sqrt question

[Han]

(04-06-2017 04:51 PM)KeithB Wrote:  pier4r:

But we all know that sqrt(36) = sqrt(4*9) = sqrt(4) * sqrt(9), right?

So the reverse is true: sqrt(-4) * sqrt(-9) = sqrt(-4*-9) = sqrt(36). This, of course, is the only way to do it in the real numbers and get an answer.

You are mixing mathematical rules, and you are also ignoring order of operations.

1. \( \sqrt{a} \) is usually only defined for \( a \ge 0 \). The multiplicative property \( \sqrt{a}\cdot \sqrt{b} = \sqrt{a\cdot b} \) likewise only applies to non-negative values of \( a \) and \(b \). Another way of putting it is that this property is a result of the assumption that the domain of \( \sqrt{x} \) is the set of non-negative real numbers.

2. The order of operations dictate that exponents are evaluated first. So \( \sqrt{-4} \cdot \sqrt{-9} = -6 \) -- however this assumes that we are not using the \( \sqrt{x} \) function as described in #1. In fact, we have have implicitly assumed that the \( \sqrt{\quad } \) operator is defined for negative values, and that the evaluation is \( i\cdot \sqrt{|x|} \)