Very simple math question

[AFB]

Why is

((-5)^.5) * ((-5)^.5) == -5

But when you multiply under the radical

(-5 * -5)^.5 == (25)^1/2 == 5 (Yes, can also be -5)



This is what I was told, no idea why.

 

[notfred]

you can't take the square root of a negative number.

 

[Juice Box]

Originally posted by: notfred
you can't take the square root of a negative number.

my thoughts exactly

 

[her209]

because the sqrt(-n) is an imaginary number.

 

[AFB]

Originally posted by: notfred
you can't take the square root of a negative number.

Yes, I know. You actually bring up a better example.

Why is

(i)^2 == -1

Because i = (-1)^1/2

Well, normally to simplify a set of radicals of the same powere that are being multiplied, you multiply under the radical.

e.g., (5)^.5 * (125)^.5 == (5 * 125)^.5 == (625)^.5 = 25

So, wouldn't two negitives cancel out when you multiply them?

(-5)^.5 * (-125)^.5 == (-5 * -125)^.5 == (625)^.5 = 25

Or in our case

(-1)^.5 * (-1)^.5 == (-1 * -1)^.5 == (1)^.5 = 1

 

[JonnyBlaze]

nerds!

(im just jealous)

JB

 

[elbosco]

(-5)^½ * (-5)^½ = (-1)^½ * (5)^½ * (-1)^½ * (5)^½

(-1)^½ = ?

? * ? = -1 and (5)^½ * (5)^½ = 5

-1 * 5 = -5

 

[her209]

Originally posted by: amdfanboy
(-1)^.5 * (-1)^.5 == (-1 * -1)^.5 == (1)^.5 = 1

I don't think that's correct.

(-1)^.5 * (-1)^.5 = (-1)^(.5+.5) = -1

 

[Soccer55]

Originally posted by: amdfanboy

Originally posted by: notfred
you can't take the square root of a negative number.

Yes, I know. You actually bring up a better example.

Why is

(i)^2 == -1

Because i = (-1)^1/2

Well, normally to simplify a set of radicals of the same powere that are being multiplied, you multiply under the radical.

e.g., (5)^.5 * (125)^.5 == (5 * 125)^.5 == (625)^.5 = 25

So, wouldn't two negitives cancel out when you multiply them?

(-5)^.5 * (-125)^.5 == (-5 * -125)^.5 == (625)^.5 = 25

Or in our case

(-1)^.5 * (-1)^.5 == (-1 * -1)^.5 == (1)^.5 = 1

But what are you actually doing when you take the square root of a number? You're looking for a number (or numbers) that, when squared, is (are) equal to the number under the radical.

For example, sqrt(25) = +5 and -5 because (-5)^2 = 25 and 5^2=25. I think what you're having problems with is that you're only looking at the principle (positive) root. Not looking at the negative solution leads you to the seeming contradiction that you've arrived at.

-Tom

 

[desteffy]

(a^.5)(b^.5)==(ab^.5) only when they are not both negative

 

[DrPizza]

Here's the reason:

Sqrt(a)*sqrt(b) = sqrt(a*b) If and only if a and b >= 0

in your case, since a and b are less than 0, you can't multiply them together.


edit: Since you typically learn operations with radicals a year (or more) before you learn about imaginary and complex numbers, teachers will often leave that if and only if part out of it when defining the operations with square roots.

edit edit:

Another weird example while we're at it..
In your earlier math text, it probably says that x^2+7 cannot be factored.


However, now that you know about imaginary numbers and irrational numbers,
x^2+7 can be factored as:
(x + i*sqrt(7) ) ( x - i*sqrt(7) )