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B Square root of y^6

  1. Aug 29, 2016 #1
    I'm trying to decide if simplifying sqrt(y^6) requires use of the absolute value bars. For example, the rule "nth root(u^n) = abs(u) when n is even" can be used to simplify sqrt(y^6) as sqrt[(y^3)^2]=abs(y^3). However, the rules of rational exponents can also be used to simplify sqrt(y^6) as (y^6)^1/2=y^(6/2)=y^3. So are the absolute value bars necessary when simplifying sqrt(y^6), or not?
    I know that the domain of sqrt(y^6) is [0,inf), so does this allow for the absolute value bars to be unnecessary?
     
  2. jcsd
  3. Aug 29, 2016 #2

    Mark44

    Staff: Mentor

    The absolute values are necessary, for the same reason that ##\sqrt{x^2} = |x|##.
    For your problem, if you wrote ##\sqrt{y^6} = y^3##, the left side is always nonnegative, for any real y, but the right side can be negative when y is negative.
     
  4. Aug 29, 2016 #3

    Ssnow

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    Gold Member

    If you specify that ##y\geq 0##, then ##\sqrt{y^6}=y^3## is correct ...
     
  5. Aug 29, 2016 #4

    micromass

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    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Here you are using the property ##(a^b)^c = a^{bc}## which is true only for positive ##a##.
     
  6. Oct 12, 2016 #5
    The absolute value bars really become necessary when we have variables under the √ sign.
    ##\sqrt{a^2}\;\;=##|a|
    Similarly, ##\sqrt{y^6}##= |##y^3##|
    But we want to make sure that the value under root is positive,
    if y=2 then
    ##\sqrt{y^6}= +y^3 \;\; or\;\;\sqrt{2^6}=+8##
    But when we have equation such as
    ##x^2=23+2##
    ##x^2=25##
    ##x=\pm\sqrt{25}##
    Then we get,
    ##x=\pm 5##

    I hope it' ll help.
     
    Last edited: Oct 12, 2016
  7. Oct 12, 2016 #6

    pwsnafu

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    Science Advisor

    This is false. ##x^2 = 25 \implies x=\pm\sqrt{25} \implies x = \pm 5##.
    ##\sqrt{25} = 5##, never -5.
     
  8. Oct 12, 2016 #7
    yes, i know. I have misstekenly typed, btw thanks
     
  9. Oct 12, 2016 #8

    Mark44

    Staff: Mentor

    The quote from @pwsnafu threw me off for a bit. At first I thought that "This is false" referred to the implication that immediately followed what he wrote. To be clear, "This is false" refers to these two lines that Deepak wrote:
     
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