# Why are the values of the variables different when the radicals are both negatives?I'm in pre-calc 11 and we're learning about radicals right now. I'm confused with the negative radicals. For...

Why are the values of the variables different when the radicals are both negatives?

I'm in pre-calc 11 and we're learning about radicals right now. I'm confused with the negative radicals. For example, why does 4w/5 - 6(2w)^1/2 have a restriction of w>0, while -(45n^2)^1/2 has a restriction of n<0. Both of them have a negative radical so why is the value of the variable different?

beckden | Certified Educator

The first (4w/5) - 6(2w)^(1/2) has a restriction that w>0 because the square root (which is the same as ^1/2 power) is restricted to positive values.

The second -(45n^2)^1/2 does not have a restriction.  45n^2 is always positive no matter what the value of n, so any value of n will always produce a positive number. The value of this expression is always negative, but that is not what you asked.

Neither has a negative radical, they have rational radicals.  If the second was (45n^2)^(-1/2) the restriction would be that n could not equal zero.  Because ^(-1/2) power is the same as 1/sqrt(n^2) and division by zero is not defined.