Wikipedia:Reference desk/Archives/Mathematics/2014 September 27

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September 27

Decimal exponents

[I really really ought to remember this!]

How do decimal exponents work? How do I calculate the result of x × 10^y.z (with a numeral in place of each variable), for example? To my surprise, decimal exponent redirects to scientific notation, which didn't even address the issue as far as I could tell, and the fractional "Rational exponents" section of exponentiation didn't either, unless it went over my head. Not trying to understand the theoretical basis for it — I just want to know how to solve x × 10^y.z, and I don't even know what to call it, since "decimal exponent" apparently means something completely different. Nyttend (talk) 11:40, 27 September 2014 (UTC)

If you have $a^{x}$ and x is not a whole number, the decimal form for x isn't helpful in the computation. Write the exponent as a fraction x = n/d, and then

$a^{x}=a^{n/d}=(a^{n})^{1/d}={\sqrt[{d}]{a^{n}}}=({\sqrt[{d}]{a}})^{n}$,

So for example $4^{1.5}$ is $4^{3/2}=({\sqrt {4}})^{3}=2^{3}=8$. Staecker (talk) 12:22, 27 September 2014 (UTC)

For one thing, I'd forgotten that x^½ is equal to the square root of x. But what to do about a fraction that's not conveniently .5? I don't know how to interpret the "d" outside the √ following your third equals sign — I know it's the denominator, but what's the significance of a number in that position? It's been several years since I took a maths course. Nyttend (talk) 12:39, 27 September 2014 (UTC)

Take 100^0.1. I just discovered that the result is the tenth root of 100. But the result of 100^0.4 is not 4x the tenth root of 100, and it's not exactly what you get when you raise the tenth root of 100 to the fourth power. I can't yet follow your instructions to use the fractional method, since I don't understand what you're doing in the <math> section. Nyttend (talk) 12:46, 27 September 2014 (UTC)

Any difference between your calculation of 100^0.4 and the fourth power of 100^0.1 will just be rounding errors, because they are the same by definition. In the fraction n/d above, n is the nth power and d is the dth root (for example if d is 3 then you want the cube root). You can do the calculations in either order. Try working out 10281024 to the power of 0.7 (=7/10) It is easiest to find the tenth root first (a whole number) then raise this to the seventh power. You should get 128. Dbfirs 13:01, 27 September 2014 (UTC)

Okay, now I think I remember better; thank you. What I needed was the simple examples such as 1028^0.1 and 1028^0.7. I was using Windows Calculator (Windows 8), typing 100, hitting the x^y button, typing 0.1, hitting enter, squaring the result twice, and adding this to memory. Next, I typed 100, hit the x^y button, typed 0.4, hit enter, and subtracted it from memory; there was a difference, with the latter method getting a result smaller by approximately 4.958 × 10^-37. (I do understand negative exponents). But then, I know quite well that 2 is the tenth root of 1028, but typing 1028, hitting the x^y button, typing 0.1, and hitting enter produces a result of 2.0007798800968566238308934329587. Is it a calculator error? If my experiment with 0.1⁴ and 0.4 had produced equal results, I would have understood much more easily. Nyttend (talk) 13:16, 27 September 2014 (UTC)

You're confusing 1024 with 1028. Try 2 [x^y] 10 to get 1024. Then 1024 [x^y] .1 to get 2 again. Generally arithmetic errors in computers, even in Microsoft products, are so rare as to be practically nonexistent. There was the infamous Pentium FDIV bug, but that's the exception rather than the rule. Not to say you don't have to watch out for rounding errors which are a different type of thing, --RDBury (talk) 15:32, 27 September 2014 (UTC)

Oops! that was my fault. Memory problems of the non-electronic variety! Dbfirs 16:37, 27 September 2014 (UTC)

My Mac calculator program gives the exact value '2' when you raise 1024 to the 0.1 power. 'Decimal exponents' isn't the best search term to use. Fractional exponents would be better. More background on fractional exponents is given in our logarithm article. When x = 10^y, x is the logarithm of y, y is the antilogarithm of x. EdJohnston (talk) 16:00, 27 September 2014 (UTC)

One has to be a bit careful when applying this to complex numbers since the complex logarithm is multi-valued. For example the complex square root can have multiple meanings depending on context.--Jasper Deng (talk) 16:42, 27 September 2014 (UTC)

the result of 100^0.4 is not 4x the tenth root of 100 -- nor should it be. Unless I'm reading the comment above wrong, there seems to be a mix-up. By the product rule of exponents,100^0.4 is equal to (100^0.1)⁴ which is the same as (100⁴)^0.1 -- and none of these is 4 times 100^0.1 . El duderino ^(abides) 10:35, 1 October 2014 (UTC)