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Complex Radical or Complex Root?

  1. Sep 25, 2014 #1
    solve a radical expression:

    1.5r250 (*r denotes the radical symbol)

    in a radical expression where the index of the root is 1.5 and the number is 250

    Use Calculator, Show Steps

    2. Relevant equations



    3. The attempt at a solution
    First of all, using a calculator the answer comes out to 39.685, but I can't really show the steps since the calculator has an XsqrtY button and simply shows the answer. I attempted to find the steps as below, I know this is wrong since I took the square after breaking down the 250....please help.

    1.5 r 250 = 1.5 r 10 * 25
    1.5 * 5 r 10
    7.5 r 10
    7.5 * 3.162
    23.715 (my answer) is not 39.685 (Scientific calculator answer)

    I can't figure out the steps to solve this without the calculator, and I can't come up with the right answer. How do you solve a radical expression where the index is not 2 (square) or 3(cube), but 1.5 instead?
     
  2. jcsd
  3. Sep 25, 2014 #2

    SteamKing

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    Are you familiar with exponents? How would you write the square root of 105 without using a radical but instead using an expression of the form 105^n. What value of n is required to express the square root of 105?
     
  4. Sep 25, 2014 #3

    LCKurtz

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    To do it by hand, write ##1.5 = \frac 3 2##, use ##a^{\frac 3 2} = \sqrt{a^3}## and simplify.

    [Edit:] Should be ##a^{\frac 2 3} =\sqrt[3]{a^2}##
     
    Last edited: Sep 25, 2014
  5. Sep 25, 2014 #4
    250^1/(3/2) = 250^2/3 = 39.685

    Thanks for the direction!
     
  6. Sep 25, 2014 #5

    250^1/(3/2) = 250^2/3
     
  7. Sep 25, 2014 #6

    Mark44

    Staff: Mentor

    This is completely unrelated to what SteamKing asked.
     
  8. Sep 25, 2014 #7

    Ray Vickson

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    What you do depends on what you mean. Do mean (1) ## x = \sqrt[1.5]{250}##, or (2) ##x = 250^{1.5}##? If you mean (1), you want to solve for ##x## that gives you ##x^{1.5} = 250##, and the solution is ##x = 250^{1/1.5} = 250^{2/3},## which you can compute using logarithms (or by pressing a button if you have a calculator with a ##x^n## button for non-integer values of ##n##). If you mean (2), you can either use logarithms, or else use the fact that 1.5 = 3/2 to write either ##x = \sqrt{250^3}## or ##x = \sqrt{250}^3##; both formulas will give the same answer. Again, an alternative would be to use an ##x^n## button on a sufficiently powerful scientific calculator.
     
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