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How do you rationalize a demoninator if the denominator is a cube root

  1. Aug 14, 2013 #1
    Hi, I know how to rationalize a denominator when it is a square root monomial or a square root binomial (through conjugation).

    For example, for a square-root monomial:

    5/√25 =

    [5(√25)]/
    [(√25)(√25)] =

    [5(√25)]/25 =

    (√25)/5 =

    1 or -1


    and, for a square-root binomial:

    5/(5 + √25) =

    5(5 - √25)/
    [(5 + √25)(5 - √25)] =

    [5(5 - √25)]/
    25 -25 =

    0/0 [undefined] or 50/0 [undefined]


    But what if the denominator is a cube root:

    x/(3√11) ?

    How do you simplify this (assuming the denominator isn't a perfect cube)? I didn't get a question similar to this correct on my assessment test.

    Thanks!
     
    Last edited: Aug 14, 2013
  2. jcsd
  3. Aug 14, 2013 #2

    symbolipoint

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    The example cube root you ask for seem not tricky because 27 already is a cube. The cubed root of 27 is 3, so the denominator is already rationalized.

    How about something like this: [itex]\frac{x}{\sqrt[3]{28}}[/itex]
    Multiply by 1 in the form of [itex]\frac{\sqrt[3]{28}}{\sqrt[3]{28}}[/itex]

    Result is [itex]\frac{x\sqrt[3]{28}}{28}[/itex]


    EDIT: That was already posted in this same posting but already responded to, when I realize now I made a big mistake.


    This example is for a cube root. With the (1/(28)^(1/3)) denominator, I should have shown multiplying numerator and denominator this way:
    [itex]\frac{\sqrt[3]{28}}{\sqrt[3]{28}}\cdot\frac{\sqrt[3]{28}}{\sqrt[3]{28}}[/itex]
     
    Last edited: Aug 14, 2013
  4. Aug 14, 2013 #3
    But doesn't

    (3√28) x (3√28) = 3√282) ??

    [because 281/3 + 281/3 = 282/3,
    because of the rule (xa) (xb) = xa+b ]

    Also, I edited my original post to eliminate the perfect cube in the denominator.
     
    Last edited: Aug 14, 2013
  5. Aug 14, 2013 #4

    CAF123

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    Find a value of ##n## in $$\frac{x}{11^{1/3}} \cdot \frac{11^n}{11^n}$$ such that the denominator is rational.
     
  6. Aug 14, 2013 #5
    That helps a lot!

    What if I multiply by: 112/3/112/3

    So the denominator will be (111/3)(112/3) = 11 !

    So the final answer will be (x3√112)/11
     
  7. Aug 14, 2013 #6

    CAF123

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    Correct.
     
  8. Aug 14, 2013 #7
    Thanks! I finally got something right :tongue2:
     
  9. Aug 14, 2013 #8

    symbolipoint

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    mileena,
    please recheck my post. I found my mistake and added better information. We square a square root to bring back the number under the radical. We cube a cubed root to bring back the number under the radical.
     
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