Simplifying radicals Q

i don’t know if this question will apply to all classes, my students are definitely not honors. In Alg1, my students had already seen simplifying radicals in Math8. They were familiar with the factor tree method and circling the doubles.

The first question I asked them, before we actually tried the “procedure”, was:

Why is sqrt(15) simplified and sqrt(12) not?

Blank stares. Then someone said because there aren’t any double to circle. I asked why we care about doubles. Nothing. They had no clue except their teacher told them to circle doubles.

So we talked about perfect squares being hidden inside of that radicand and that’s why we can simplify it further. The “doubles” just find perfect squares.

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3 thoughts on “Simplifying radicals Q

  1. I think a key note is that: sqrt(ab) = sqrt(a)sqrt(b).

    In this way, we can write something like:
    sqrt(12) = sqrt(4×3) = sqrt(4)sqrt(3) = 2sqrt(3),
    where we are using ‘sqrt’ to denote the positive square root.

    But: Are students aware of this (multiplicative) property?
    Do they see how it is relevant? (e.g., in the example worked out above)
    Can they explain why it is true? (I’m not sure if I can…)

    In general: (ab)^n = a^n b^n … and the “sqrt” case is for n = 1/2.
    But is there a reason to believe a priori that this rule of exponents would hold not just for whole numbers, but for rational numbers more generally?

    It is not tough to see why it holds for whole number exponents.
    For example, we can show why (ab)^3 = a^3 b^3:
    (ab)^3 = ababab = aaabbb = a^3 b^3; all we need is commutativity, associativity, and patience.

    But what does (ab)^(1/2) even mean?
    (Sorry for all the questions and not much by way of answers!)

    Like

    • Yes interesting line of thought. My student don’t know yet that sqrt (ab) equal the sqrt of a times sqrt of b. They only have in their heads a memorized procedure from last year.
      They also dont know that sqrt is exp of 1/2!
      So much wonderful math them to dig into but so many standards to get to….

      Like

      • What does an explanation of sqrt(12) = 2sqrt(3) look like without appealing to the aforementioned fact?

        As you write:
        >So we talked about perfect squares being hidden inside of that radicand and that’s why we can simplify it further. The “doubles” just find perfect squares.

        Hence my (genuine!) curiosity:
        How does this work in practice without using sqrt(ab) = sqrt(a)sqrt(b)?

        Like

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