Square root of a square

[jpanknin]

I'm having trouble with this expression [imath]\sqrt{x^2}[/imath].

I was always taught that [imath]\sqrt{x^2} = x[/imath], but in class last week it was shown as [imath]\sqrt{x^2} = |x|[/imath]. Which kind of makes sense because x could be positive or negative. I still don't fully understand this, though, especially if you take the [imath]x^2[/imath] first (which would make the inside of the square root positive) and then the square root, you should get the positive value and not need to worry about absolute value (or at least this is how I'm thinking about it).

Also, the |x| doesn't match with exponent operations such as: [imath]\sqrt{x^2} = (x^2)^{1/2} = x^{2/2} = x^1 = x[/imath], but this is different from |x|.

So I get conceptually how x in [imath]\sqrt{x^2}[/imath] could be positive or negative to begin with, but having trouble extending that the |x| answer.

 

[Dr.Peterson]

Many ideas like this can be introduced in a way that oversimplifies the general facts. You have to grow out of that.

When you square a positive number and then take the square root, the result is the original number. For example, square 3 to get 9, then take the square root (which is specifically the non-negative number whose square is that number), you get 3 back.

But if you start with the negative number -3 instead, squaring it gives 9 again, but the square root is still 3, not -3. (The number doesn't remember its history and go back to where it started!)

So to make a general rule, you have to consider the sign; and taking any number and making it positive, as happens here, is called the absolute value.

Similarly, the rule that (a^m)^n = a^(mn) is true when a is positive, or when m and n are integers, but not in general. So what is true when you first meet exponents isn't true when you grow up. That's life.

 

[jpanknin]

Ok, so is there a difference in the question perhaps? Meaning, if I have [imath]\sqrt{x^2}[/imath], one question could be 1) what was the original value of x and 2) could be what is the value of the expression after it is evaluated. So in your example of the square root of 9, I can see 1) being |3|, but 2) would just be 3 (positive).

 

[Dr.Peterson]

1) what was the original value of x and 2) could be what is the value of the expression after it is evaluated.

So in your example of the square root of 9, I can see 1) being |3|, but 2) would just be 3 (positive).

I think you have it backward.

(1) The original value of x simply can't be determined from the value of the expression. But obviously it is just whatever you chose to use for x: 3 in my first example, and -3 in the second. That's not an absolute value!

(2) The value of the expression is |x|, because it is never negative. This is an absolute value: |3| = 3 and |-3| = 3.