How to workout fractional indices (without a calculator)??

komencanto

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So, 2^2 = 2 x 2 = 4
2^3 = 2 x 2 x 2 = 8
2^2.5 = ?????

Sure, you can just put it into your calculator, but what does it actually mean to put something to a fractional indice, and how could it be worked out by hand. If 2^3 (8) is 2 timed by itself twice, and 2^1.5 is 2 timed by inself .5 times??
And how can you concieve of timesing something by itself negative times, such as 2^-2 (.25)??

Say what??
 
komencanto said:
So, 2^2 = 2 x 2 = 4
2^3 = 2 x 2 x 2 = 8
2^2.5 = ?????

Sure, you can just put it into your calculator, but what does it actually mean to put something to a fractional indice, and how could it be worked out by hand. If 2^3 (8) is 2 timed by itself twice, and 2^1.5 is 2 timed by inself .5 times??
And how can you concieve of timesing something by itself negative times, such as 2^-2 (.25)??
Click to expand...

Yeah, you have to get away from thinking that an index of n means something has to be multiplied by itself n times. It's easier to think in terms of the following relations:

(2<SUP>p</SUP>)<SUP>q</SUP>=2<SUP>pq</SUP>
2<SUP>1/p</SUP>= p th root of 2
2<SUP>-p</SUP>= 1/2<SUP>p</SUP>

From the first two relations you can see that
2<SUP>2.5</SUP>=(2<SUP>5</SUP>)<SUP>1/2</SUP>
which is the square root of 32.

It's difficult to do it by hand though, since taking the square root is the tricky thing. Log tables or binomial expansion sounds about right.
 
a^(x/y) = yth root of a^x.

a^(-x) = 1/(a^x).

When we were first taught this, I think we were told to just accept it...

You could look at the second rule this way:

2^3 = 8
2^2 = 4 (i.e. half of 2^3)
2^1 = 2 (i.e. half of 2^2)
2^0 = 1 (i.e. half of 2^1)
2^(-1) = 1/2 (i.e. half of 2^0)
2^(-2) = 1/4 (i.e. half of 2^(-1))

and so on.

2^(1.5) = 2^(3/2) = 2nd root of 2^3 = Square root of 8 = 2(Sqrt(2))

clicking preview: Damn, beaten to it.
 
Depends on the quality (and type) of mathematical education you received.

"In my day" (which is around the time that dinosaurs roamed the earth) binomial expansion and logarithms were done at the age of 13-14 - I don't know how this corresponds to year 11.

This page is a little clearer in terms of its examples and explanation

http://www.ucl.ac.uk/Mathematics/geomath/level2/series/ser8.html#l5


2 ^ 2.5 = 2 x 2 x 2^0.5

2 ^ 0.5 is the square root of 2 (= 1.4 ish)

so 2 ^ 2.5 = 2 x 2 x 1.4 ish = 5.6 ish


From the binomial......

(1+x) ^ n = etc

x=1, n = 5/2


= 1 + 5/2 + 15/8 + 15/48 - 15/384
= 1 + 2.5 + 1.875 + 0.3125 - 0.03906 = 5.6484375


Which is just an approximation
 
The Don said:
"In my day" (which is around the time that dinosaurs roamed the earth) binomial expansion and logarithms were done at the age of 13-14 - I don't know how this corresponds to year 11.
Click to expand...
Oh dear. I'm in S6 (17-18yo) and I've only just done binomials. Logarithms we did a year ago.

Of course, now we have Supa-Dupa Do-Everything Calculators™. One of my friends has a calculator that solves systems of equations for him. I mean, that's not on. He should suffer like the rest of us.
 
Mendor said:
Oh dear. I'm in S6 (17-18yo) and I've only just done binomials. Logarithms we did a year ago.

Of course, now we have Supa-Dupa Do-Everything Calculators™. One of my friends has a calculator that solves systems of equations for him. I mean, that's not on. He should suffer like the rest of us.
Click to expand...

Exactly, I mean if you didn't learn how to solve these things, you wouldn't be able to apply the solution in everyday solutions (i.e. never) :D

I suspect things are dofferent since I went to school. We only had the odd numbers, even numbers hadn't been invented
 
The Don said:
I suspect things are dofferent since I went to school. We only had the odd numbers, even numbers hadn't been invented
Click to expand...
"We only had the odd numbers, AND WE LIKED IT!!"
 
(Grabs trusty Faber-Castell SLIDERULE off shelf.)

2 ..to the...1.5 is (squints over top of spectacles)..about 2.83??

Finds casio calculator. Battery dead, won't switch on, blast,
(Hits run "calc")

2.82427etc.

So the old sliderule still works. All I need is a magnifying glass.
 
komencanto said:
So, 2^2 = 2 x 2 = 4
2^3 = 2 x 2 x 2 = 8
2^2.5 = ?????

Sure, you can just put it into your calculator, but what does it actually mean to put something to a fractional indice, and how could it be worked out by hand. If 2^3 (8) is 2 timed by itself twice, and 2^1.5 is 2 timed by inself .5 times??
And how can you concieve of timesing something by itself negative times, such as 2^-2 (.25)??

Say what??
Click to expand...

Try this approach, it helps me to keep the conventions straight.
 
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