Square roots of positive numbers

[bluskies]

Homework Statement

If a and b are positive real numbers, and \lambda^{2} = ab, then \lambda = \pm \sqrt{ab}.

Homework Equations

None.

The Attempt at a Solution

This is more of a conceptual question that has always escaped me. I do not understand how the square root of two positive numbers could possibly be negative. Since a and b are positive, how can there be any negatives in the square root of their product? Any guidance on this subject would be very much appreciated.

 

[rock.freak667]

(-1)² =1

so squaring -√(ab) will give (ab) regardless of a + or - before the square root.

 

[dacruick]

I think you might be looking too hard into this. There are no negatives in the square root. the negative is outside it. say both a and b are 5. then lambda² = 25. lambda therefore is equal to positive root(25) or negative root(25). Or in other terms, lambda² = 5² or (-5)²

 

[Fightfish]

A numerical example can illustrate this. Let a = 1 and b = 4 for instance. Then we have \lambda^{2} = 4. We realize however, that because negatives cancel upon multiplication, in fact (-2)^2 = 2^(2) = 4, and so both -2 and 2 are possible solutions.

 

[bluskies]

Thank y'all for your help, that makes sense now. :)