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Square root of x squared is…

June 18, 2010 am30 12:20 am
Commenters have convinced me that my objection was over-picky, 
and I am backing off my contention - jd]

I think New York State doesn’t know. That would explain their answer to the Integrated Algebra 2 and Trigonometry Regents #32. They got it wrong…

Let’s look.

We want to simplify \sqrt{x^2}

Simple, right? That should be x.

Let’s test.

x x^2 \sqrt{x^2}
1 1 1
0 0 0
-1 1 1

Look at that! When x is non-negative we get x, but when x is negative we get -x (in other words, the opposite, a positive number)

Looks like \sqrt{x^2} = |x| … surprised?

Now look at NY State’s problem: Express 5\sqrt{3x^2} - 2\sqrt{27x^3} in simplest radical form.
See the trap. Watch NYSED fall into it…
Really, few of us teach students about this detail. And you’ll survive. But we should be telling you.

8 Comments leave one →
  1. Noel permalink
    June 18, 2010 am30 12:34 am 12:34 am

    Before you always start a problem you state the domain restrictions, and based on that problem x must be greater than or equal to 0.

    So even though it is true

    sqrt (x^2) = |x| , because obviously the answer will always be positive, because x^2 will always yield a positive number, and taking principle square root of a positive number would yield an positive answer.

    So if x were let’s say -3, then it’s true sqrt ((-3)^2) = -3 would not hold true looking at principle square root value.

    But since we cannot even have a negative value for x, because if we do the original expression would not work in the real number set, I don’t think the |x| is necessary, I think it is implied we are dealing with only positive values of x.

  2. June 18, 2010 am30 12:56 am 12:56 am

    Before you always start a problem you state the domain restrictions…

    But there are no restrictions stated. And this is an exam where they freely move between the reals and the complex. No assumptions are warranted.

    By the way, when I type (or you can too)

    ($)latex \sqrt{x^2}($) and take away the open and closed parentheses, you get

    Other easy, useful layout: \frac{num}{den}

    So ($)latex \frac{sin\theta}{cos\theta} = tan\theta($) becomes:
    \frac{sin\theta}{cos\theta} = tan\theta


    • Noel permalink
      June 18, 2010 am30 1:06 am 1:06 am

      I guess you are correct the state should have looked over the exam effciently. I doubt that a lot of people even remembered the |x|, it is a 3 hour exam, in which you have to account many questions, I don’t think many students would stress or worry about details. It was 2 point credit worth option, so if the state were to take off for |x|, they could not take off much, but like you said according to state the answer was the answer without the absolute value of x. The mathematical symbols you use by that method are effective, but also very small some of the time, and hard to see, so I find it better to type it regularly.

      Back on topic, I guess this time you can blame the people in Albany, although they are humans, and humans make errors, but they are being paid a lot to do this correctly, without any errors, but I guess errors will always occur.

    • June 18, 2010 am30 5:35 am 5:35 am

      To help Noel out a bit… jd is also making an assumption on the domain. The formula $\sqrt{x^2} = |x|$ holds for all real numbers $x$ (Exercise: Check what happens over the complex numbers).

      If the exam freely moves between the reals and the complexes then a typical student/NYSED probably doesn’t have enough experience to know mathematical conventions and assumptions often held – that is, in a question like “express in simplest radical form” using a variable of x, it is safe to assume we are working over the reals (not the set of positive/nonnegative numbers or complex numbers).

      I’ll also note that there is often a lot of confusion between what “finding square roots” means versus $\sqrt{x}$. Normally, “the square root” is often used to refer to the principal square root and denoted by $\sqrt{x}$. If you’re interested, see for more information regarding this and the square root of complex numbers.

      • June 18, 2010 am30 6:59 am 6:59 am

        It’s a fair point. But these guys (the ones writing the test) are supposed to be smart about it. In #6 they call for an imaginary number, in #18 they assume the domain is the Reals, in #11 it doesn’t matter.

        That being said, I’ll back down on this one. But they would have been better restricting x.

  3. June 18, 2010 am30 1:05 am 1:05 am

    sqrt(x^2) is one of my standard test equations to see how good a piece of graphing software is.

  4. Anonymous permalink
    June 18, 2010 am30 10:07 am 10:07 am

    sorry to interrupt: but anyone want to discuss the geometry regents?

  5. Mike permalink
    June 18, 2010 am30 11:32 am 11:32 am

    I’m sorry, but I don’t understand anything you guys are saying on this page.

    is there any way you could dumb down your logic?

    (I understand that you’re suggesting that a certain Q was unfairly or unclearly expressed, but is there any way you could dumb it down here? I want to e-mail this page to my math teacher, but first I’d like to understand what it is that you’re talking about.)

    So, please help.

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