Intuitive explanation of fractional exponents?

[musicgold]

Homework Statement

What would have caused humans to come up with fractional exponent notations?

Homework Equations

The Attempt at a Solution

I understand that it makes sense to use the exponent notation when we have to multiply the same number a number of times. For example, 10^8 is the short form for writing 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10.

But, I am not able to imagine a scenario where a human ancestor had to use this notation: 10^0.8. Is there an intuitive way to understand this?

Thanks.

 

[tnich]

Homework Statement

What would have caused humans to come up with fractional exponent notations?

Homework Equations

The Attempt at a Solution

I understand that it makes sense to use the exponent notation when we have to multiply the same number a number of times. For example, 10^8 is the short form for writing 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10.

But, I am not able to imagine a scenario where a human ancestor had to use this notation: 10^0.8. Is there an intuitive way to understand this?

Thanks.

Fractional exponents pop up in compound interest calculations.

 

[phyzguy]

It's really quite simple. If you start with the concept that 10X10X10 = 10^3, you soon see that when you multiply two numbers, you add the exponents. So then you ask, why should the exponents only be integers? Why not 10^x * 10^x = 10^2x = 10 = 10^1? so x = 1/2. So 10^1/2 = sqrt(10). From there you can extend the idea to any real number, which gets you to the concept of logarithms. There are other reasons why this makes sense, but that's the basic idea.

 

[musicgold]

Thanks.

But I am not talking in abstract terms. In terms of physical objects, what does 10^0.8 = 6.31 mean?

For example, I can tell a kid that all 10^8 means is multiplying 10 eight times. How do I explain 10^0.8 = 6.31 to her?
How are the 10 and 6.31 related in the physical world?

I hope I am making sense. Thanks again.

 

[Ray Vickson]

Thanks.

But I am not talking in abstract terms. In terms of physical objects, what does 10^0.8 = 6.31 mean?

For example, I can tell a kid that all 10^8 means is multiplying 10 eight times. How do I explain 10^0.8 = 6.31 to her?
How are the 10 and 6.31 related in the physical world?

I hope I am making sense. Thanks again.

(1) If at time $t=0$ (measured in years) you have $\$1$ in a bank account that pays 12% per annum, but with interest compounded daily, your account contains $\$1.12$ at the end of the year ($t = 1$). What would it contain at time $t = .75$ (3/4 of a year)? What would it contain at $t = 2.5$ (2 and 1/2 years)? The answers are $\$1.12^{.75}$ and $\$1.12^{2.5}$---so fractional exponents!
(2) Suppose you have a radioactive substance with a half-life of 2 years (so after 2 years, half of the radioactive substance is gone---converted to something else). What fraction of the initial amount would be left at $t = 0.9$ years? At $t = 11.66$ years? Again, these questions/answers involve fractional exponents.

Of course, if you go back far enough our ancestors did not know about compound interest or radioactivity, and so might not have needed fractional exponents.

As to how people came up with these things: who knows? How did Copernicus come up with the current view of the solar system? How people come to realize the Earth is round? How did Newton and Leibnitz come to invent calculus? They all did, and we benefit from that.

 

[PeroK]

Thanks.

But I am not talking in abstract terms. In terms of physical objects, what does 10^0.8 = 6.31 mean?

For example, I can tell a kid that all 10^8 means is multiplying 10 eight times. How do I explain 10^0.8 = 6.31 to her?
How are the 10 and 6.31 related in the physical world?

I hope I am making sense. Thanks again.

I would say the answer to your question lies less in the arithmetic properties of exponents than in the more general mathematical properties. For example, the exponential function $e^x$ is one of the most useful across mathematics, pure and applied. To define this function for all real numbers $x$, you must deal with real exponents, both rational and irrational.

I don't think equations like $10^8 = 6.31$ are particularly useful in their own right, but:

$\frac{d}{dx}e^x = e^x$

is more than extremely useful.

 

[CWatters]

I think 0.8 is a bad example to start with. Makes more sense to explain

1, 2, 3 are powers (squared, cubed etc) to
1/2 1/3, 1/4 are roots (square root, cube root etc)

So its best to express 0.8 as 8/10 eg X to the 8th power and 10th root.

See also

https://medium.com/i-math/what-do-fractional-exponents-mean-1bb9bd2fa9a8

 

[Mark44]

So its best to express 0.8 as 8/10 eg X to the 8th power and 10th root.

Or in terms of radicals:
$$10^{0.8} = \sqrt[10]{10^8}$$
which is the same as $$\sqrt[5]{10^4}$$

Something in the same vein, but maybe easier to comprehend would be $8^{4/3} = (\sqrt[3] 8)^4$, which can also be written as $\sqrt[3]{8^4}$.
In either form, when completely simplified, this would be $2^4 = 16$
Any positive number to a rational power (and some negative numbers) can be rewritten in terms of radicals. For example, $x^{m/n} = \sqrt[n]{x^m}$ or $(\sqrt[n] x)^m$, take your pick. Here a and b are positive integers, but the equation may or may not hold for negative real x, depending on the power.

 

[Mark44]

But, I am not able to imagine a scenario where a human ancestor had to use this notation: 10^0.8.

I can imagine such a scenario.
The keys on a piano are arranged in several octaves, with A below middle C being about in the middle of the keyboard, with a frequency of 440 Hz. The next higher A is an octave higher in pitch, and its frequency is double that of A below middle C. The way the tones are laid out, by long tradition, is that each semitone increases in frequency by a factor of $2^{1/12}$ from the previous tone. So A frequency = 440 = $440 \times 2^0$ Hz, A# frequency = $440 \times 2^{1/12}$ = 466 Hz, B frequency = $440 \times 2^{2/12}$ = 494 Hz, $\dots$, next A frequency = $440 \times 2^{12/12}$ = 880 Hz, and so on.

I'm not sure how long this has been known, but I would guess something like at least 150 years.