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Fractional Exponents

Key Questions

  • The reciprocal of the number associated with the radical is the power needed.

    Examples:

    #root3(5)=5^(1/3)#
    #root7(2)=2^(1/7)#

    If the radicand (number under the radical sign) has a power in it, the same method still works:

    #root4(9^2)=9^(2/4)#

    This can be simplified to get #9^(1/2)#.

  • #x^(a/b) =rootb(x^a) = (rootb(x))^a#

    You can just remember this rule, or you can learn about why this is:

    fractional exponent #1/b#

    So first we're going to look at an expression of the form: #x^(1/b)#.
    To investigate what this means, we need to go from #x to x^(1/b)# and then deduce something from it.

    #x^1 = x^(b/b) = x^(1/b*b)#
    What does multiplication mean? Repeated addition. So we can instead of multiplying by b, adding the number to itself #b# times.
    #x^(1/b+1/b+1/b+1/b +...)# (b times)

    There is a rule you use when multiplying numbers with the same radical: add the exponents. If we reverse this rule, we get:
    #x^(1/b)*x^(1/b)*x^(1/b)*x^(1/b)*x^(1/b)...# (b times)

    Now, we still know that this number is equal to #x#. So now we have to think a bit. What number, multiplied by itself b times, gives you #x#.
    It's the bth-root of #x# => #x^(1/b)=rootbx#

    For example: #8^(1/3)#
    If we multiply this by itself 3 times we get:
    #8^(1/3)*8^(1/3)*8^(1/3) = 8^(3/3) = 8#
    What number multiplied by itself 3 times, gives you 8.
    It's of course #root3(8) = 2#

    What about #a/b#?
    To know what #x^(a/b)# means, we can further rely on our previous findings:
    #x^(a/b) = x^(a*1/b) = x^(1/b+1/b+1/b+1/b...) # (a times)
    #= x^(1/b)*x^(1/b)*x^(1/b)...# (a times)

    Repeated multiplication is equal to exponentiation, so we can write:
    #= (x^(1/b))^a = (rootbx)^a#

    You can also bring the exponent in the root:
    #= rootb(x^a)#

  • We can rewrite:

    #b^{m/n}=root{n}{b^m}#


    Example

    #3^{5/7}=root{7}{3^5}#


    I hope that this was helpful.

  • I will show you what fractional exponents are

    Suppose we are asked to simplify this :

    #(16)^(1/4)#

    Basically this means that we have to find the #4^(th)# root of 16

    so in the form of a picture it will be like thisenter image source here

Questions