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Raising a number to a non-integer power.

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Circlotron:
Okay, so 4^3 = 4x4x4 = 64.
But 4^3.9 = ?? = 222.861.
How can you multiply a number by itself a non-integer amount of times?

greenpossum:
It's just the extension in the real number domain of taking the exponent. Mathematics is like that, analogies can only go so far.

But if you like, think of it as 4^(39/10) so take the 39th power of 4 and then the 10th root. Things get tricky to imagine when the exponent is irrational.

Ian.M:
I'm no mathematician so the terminology here is going to give a real mathematician fits.  Without getting into logarithms, or huge powers of four, break that down to:
 43 x root10(4)9

root10(n) is the tenth root of n, i.e, the function where if:
 m=root10(n)
then:
 m10=n

That approach is valid for any rational real number.

As GreenPossum points out, its not any help with irrational numbers. so letting X be the exponent, although it gives you a function with an infinite number of points along any interval of the X axis, it isn't continuous.   To get the  'in-between' irrational values you have to consider exponentiation as multiplying the logarithm of n by the power X, then taking the antilog, which works in any base of logarithms, and is a continuous function, numerically identical to the 'classic' multiplying numbers by them-selves X times concept for any real rational X.

Where it gets gnarly to conceptualise is when you start extending exponentiation YX into the complex number domain for both Y and X.   I cant wrap my brain around the number of dimensions required to visualise that plot - its bad enough visualising the complex result of a function operating on a single real number parameter, which is a 3D plot.

RoGeorge:

--- Quote from: Circlotron on May 27, 2020, 06:00:11 am ---Okay, so 4^3 = 4x4x4 = 64.
But 4^3.9 = ?? = 222.861.
How can you multiply a number by itself a non-integer amount of times?

--- End quote ---

Short answer, the exponential function is defined as a polynomial.  Explanation starts at minute 7:00, at about 7:50 is exactly your question.

tom66:

--- Quote from: Circlotron on May 27, 2020, 06:00:11 am ---Okay, so 4^3 = 4x4x4 = 64.
But 4^3.9 = ?? = 222.861.
How can you multiply a number by itself a non-integer amount of times?

--- End quote ---

Because log10(222.861)/log10(3.9) = ~4.  And that is the division of two base10 logarithms.  You can easily define log10(n), and you have inferred what the solution to your non-integer power is with nothing more than division.

Mathematical operations often have an inverse.

Try fractional and negative factorial if you -really- want to blow your mind.

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