How come 3^(221/92) is very very nearly fourteen?

Is there supposed to be some deeper mathematical meaning in this?

12 Answers

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  • 1 month ago

    Let y=3^(221/92), then

    log(10)y=(221/92)log(10)3

    =>

    log(10)y=2.402173913(0.477121254)

    =>

    log(10)y=1.146128223

    =>

    y=10^1.146128223

    =>

    y=14.00000604~14.

  • 1 month ago

    3^(221/92) => 

    3^2.402... =>

    On your calculator 

    13.997... ~ 14

  • Amy
    Lv 7
    1 month ago

    Because you typed log base 3 of 14 into your calculator and asked it to display the result as a fraction, and the answer was 221/92.

  • 1 month ago

    221/92=2.402

    3^2=9

    3^2.402=14.000006

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  • Bill-M
    Lv 7
    1 month ago

    There is nothing special about that number.

    221 divided by 92 is not a special number. 2.4022

  • 1 month ago

    No deeper meaning. 

    221/92 is just a fairly good approximation of logarithm

    log₃(14) = log(14) / log(3) = 2.4021735027...

    which is a transcendental number

    It is always possible to find fractions that approximate transcendental numbers with increasing accuracy.

    For example

    3^(12/5) = 13.9666101652...

    more accurate is

    3^(221/92) = 14.0000063108...

    but

    3^(572719/238417)= 13.99999999993129....

    How many decimal places do you need?

  • sepia
    Lv 7
    1 month ago

    3^(221/92) = 14.00000631081314

  • 1 month ago

    Let's assume you wanted an exponent that would make it *exactly* 14:

    3^x = 14

    We could take the common log of both sides:log(3^x) = log(14)

    Then use this rule of logarithms --> log(a^b) = b log(a):

    x log(3) = log(14)

    x = log(14) / log(3)

    x ≈ 2.40217350273...

    Let's look at the value of your fraction:

    221/92 = 2.40217391304...

    You've picked a fraction that is the same out to 6 decimal places, so it makes sense that the result would be pretty close to 14.

    There is no deeper meaning other than you've gotten a decent rational approximation of log(14) / log(3).

    You can get a continued fraction form of log(14) / log(3) as:

    x = 2 + 1 / (2 + 1 / (2 + 1/(18 + ...)))

    If we stop the continued fraction after 18, we get 221/92 which is what you had.

  • David
    Lv 7
    1 month ago

    3^221/92 is the same as 3 to the power of 221/92 or 92 root of 3 to the power of 221 and they both work out as 14.00000631 or 14 rounded to nearest integer which means a whole number.

  • 1 month ago

    14.000006310813142909573588...

    coincidence. 

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