PH Algebra II - Quiz 7a

[Mrs Bee]

I was surprised to see that in the Teacher Key the variables in the answers are never inside absolute value brackets - it doesn't seem right when they are the root of something with an even index (see Theorem 7-3, p.295). Could someone clarify?
Thank you!

[Cindy in Indy]

Hello Mrs. Bee,
As you correctly state, the even root of even power of a variable should be noted with absolute value brackets. In lesson 7-1, the answers are noted correctly: e.g. #37-40.
Can you provide an example of where the absolute value bars are omitted? I will try to address that example (which very well may be noted incorrectly.)

[Mrs Bee]

NB: I am talking about the MP Quiz 7A in the separate book, not the PH text. For instance, the answer to exercise #3 doesn't have x-1 inside absolute value, nor the answer to exercise #4: shouldn't it be 5 times the absolute value of a, times the square root of 3a? Same thing with exercise #5, shouldn't it be 3 times the absolute value of x? Exercise #10, too.

I now see that the answer to exercise #2 does have the absolute value of t, so it's not true what I wrote above, that none of the answers showed the absolute value brackets - that one does.

Thank you!!

[Mrs Bee]

Maybe you don't have the Quiz book handy... For example, exercise #4 mentioned above asks to simplify by factoring the square root of 75a³. This one is the easiest to write here like this!

[Cindy in Indy]

Hi Mrs. Bee,
For that example, the reason the factored out sqrt(a^2) is not in absolute value bars, is because when it appears under the square root sign as a^3, a is required to be > 0 (because a^3 must be non-negative to have a real principal root).
The nonnegative requirement for the radicand appears in Theorem 7-2 on p.297. It is required when multiplying 2 radicals (and similar situations). This is a subtle point, and the textbook is unfortunately not always consistent with the notation. We tried to be consistent in the answer key. Please let me know if you find other examples where it appears to be incorrect, so that we can make the correction or explanation if needed.

For example on p.298, the book is not consistent for Try This n. That answer should not have y in absolute value bars, because y had to be nonnegative to apply Theorem 7-2 to simplify.

I hope this helps. It is subtle, and potentially confusing! You are doing a great job carefully thinking and reading through this topic.

[Mrs Bee]

Ah, now I see the light! So, still speaking of the quiz, exercise #2 does need the absolute value sign for t, because the original radicand would have been nonnegative even if t had been negative, since t⁴ is positive no matter what. But in exercise #4 (or #3), the variable must be positive, because if it was negative, its odd exponent would keep it negative, causing the whole radicand to be negative.

Exercise #10 in this quiz [that is, 5 over (square root of 2 + square root of x)] could be compared to exercise #51 in the PH book on p.308: the PH solution key puts a and b in absolute value brackets, but you're saying that we don't need them, because of course a and b must be nonnegative to begin with, since they are radicands. Correct? Bonus question: in exercises like this, should we worry about making explicit the conditions we need so that a denominator is not 0?

I feel like I am in a bind... how do I grade the work that shows absolute value signs where they are not needed? I'm afraid the book has privileged forming the habit of making sure we don't decide the sign of a variable (by using the absolute value brackets), over the habit of making sure a radicand is nonnegative, which could well mean a variable cannot be but positive (and so the absolute value sign is not needed.)

[Cindy in Indy]

Hi Mrs. Bee,

You are correct about quiz 7a: #2, #3, #4, #10. You've got it! Regarding your "bonus question" for #10, the textbook did not cover domain restrictions in Ch7, so we did not include them in the answer key. But you are correct, the domain should not include x = 2. You will revisit restricted domains in precalculus.
Regarding p.308, #51, showing the absolute value signs is not necessary (even though the textbook includes them). I would not mark off if the student shows the absolute value bars, though they are unnecessary. For the answers to Lesson 7-1, I would be sure to show the absolute value bars, because that is what the lesson is trying to teach. For later lessons, I would not be strict about it. On the quiz, I would require the absolute value bars for #2, but not for #3,4, or 10. I would not mark off if the student uses absolute value bars, but I would note where they are not needed.
Let me know if you have further questions - these are great!

[Mrs Bee]

Thank you so much, this was most helpful!