Mozart's Allegretto In Piano Concerto 17
Twitching & Birdwatching3 mins ago
To the power...
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This is probably childishly simple, but - as a mathophobe of long standing - I have no idea what it involves. I know that "5 to the power of 5", for example, means...5x5x5x5x5...but what does "5 to the power of 0.68" mean? Is it just 68% of 5...ie 3.4...or some quite other figure? If so, how does one obtain it? My thanks in advance.
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Hi QM - the reciprocal of powers are roots, therefore an equivalent to 5^0.68 would be the 100th root of 5^68 or even the 50th root of 5^34 i.e. 2.987442829 which is not childishly simple to calculate, although not impossible with the use of logarithms (& antilogs). Better off just poking the figures into a calculator or suchlike.
It may be easier to think of 0.68 as a fraction. Then you have 5^(68/100) or 5^(17/25). This work more easily with a simple fraction. i.e. 2/3. Here we'd have 5^(2/3) {by the way ^=to the power of ...}. Then we can treat the sum as [ (5^2) * (5^ {1/3} ) ]
5^(1/3) is the same as the cubed root of 5.
To work out the root/cubic root of a number, I believe is done using some kind of converging series. e.g. my old maths teacher proved that any number to the power of 0 is 1 (x^ 0 = 1), using some series which I now can't remember.
I shall look into this further overnight, for a more precise answer.(unless someone gets there first)
These are nasty sums and I would advise using a calculator if you must work something like this out. I would.
5^(1/3) is the same as the cubed root of 5.
To work out the root/cubic root of a number, I believe is done using some kind of converging series. e.g. my old maths teacher proved that any number to the power of 0 is 1 (x^ 0 = 1), using some series which I now can't remember.
I shall look into this further overnight, for a more precise answer.(unless someone gets there first)
These are nasty sums and I would advise using a calculator if you must work something like this out. I would.
Good grief, folks! Did you read the �mathophobe' element of my question? Reciprocals? Roots? Logarithms? Antilogs? Converging series? I have literally no idea what you are talking about. I'm not so much a mathophobe as a mathematical cretin.
I do have a simple calculator, but which figures should I �poke into it' and, having poked them, what - and how - should I ask the calculator to do with them?
I'm assuming that 2.987 etc actually is the answer to 5^0.68. (See? At least I've now learned that ^ means �to the power of'.)
Thanks for your willingness to help, however. I know how easy it is for knowledgeable people to assume they are being clear when those they mean to help are standing there, like Manuel in �Fawlty Towers', saying ��Que?'
I do have a simple calculator, but which figures should I �poke into it' and, having poked them, what - and how - should I ask the calculator to do with them?
I'm assuming that 2.987 etc actually is the answer to 5^0.68. (See? At least I've now learned that ^ means �to the power of'.)
Thanks for your willingness to help, however. I know how easy it is for knowledgeable people to assume they are being clear when those they mean to help are standing there, like Manuel in �Fawlty Towers', saying ��Que?'
(Posted in 2 parts):
OK, Quizmonster, let me have a try:
I assume that you know what a square root is. e.g. the square root of nine equals three. Now, if I want to write that sentence using mathematical symols, I can write it one of two ways. You've almost certainly seen it written as 'Tick symbol' 9 =3 but the other way to write it is "9 to the power of one half" = 3. So a power of one half means the same as 'square root'.
Ok, let's move on one step to cube roots. Since 5 cubed equals 125, then the cube root of 125 equals 5. Writing this in the form which I've just introduced you to, we get 125 to the power of one third = 5.
So, we've dealt with the meaning of powers like one half, one third and so on. (Hopefully you can now understand why 81 to the power of one quarter = 3, or 32 to the power of one fifth = 2).
OK, Quizmonster, let me have a try:
I assume that you know what a square root is. e.g. the square root of nine equals three. Now, if I want to write that sentence using mathematical symols, I can write it one of two ways. You've almost certainly seen it written as 'Tick symbol' 9 =3 but the other way to write it is "9 to the power of one half" = 3. So a power of one half means the same as 'square root'.
Ok, let's move on one step to cube roots. Since 5 cubed equals 125, then the cube root of 125 equals 5. Writing this in the form which I've just introduced you to, we get 125 to the power of one third = 5.
So, we've dealt with the meaning of powers like one half, one third and so on. (Hopefully you can now understand why 81 to the power of one quarter = 3, or 32 to the power of one fifth = 2).
(Second part):
So what about powers like two thirds? Well, 8 to the power of two thirds is a calculation which is worked out in two stages. The first part is to calculate 8 to the power of 2 (which equals 64). The second part is to work out 64 to the power of one third (which gives the answer, 4).
Let's try one more like that but where the numbers don't work out quite so easily: If we look at 5 to the power of 1.5, we can re-write 1.5 as a fraction, so we've got 5 to the power of (3 over 2). It's a two-stage calculation, like before, so the first bit is to work out 5 to the power of 3. This gives us 125. The second part is to work out 125 to the power of one half (i.e. the square root of 125). My calculator gives this as 11.180 (to 5 significant figures)
Now, back to your question: 5 to the power of 0.68 means 5 to the power of 68 hundredths. So, the first stage is to work out 5 to the power 68. That's far too big a number to type out so I'll just call the answer ' x '. So we've now got this massive number but we've still got to do the second part of the calculation. i.e. we've got to work out the hundredth root of that great big number, x. At this point, I have to cheat a bit and reach for my calculator to find an answer of 2.9874 (to 5 significant figures).
Hopefully, I've answered the first part of your question, regarding the meaning of 5 to the power of 0.68. What I haven't done is explain how mathematicians have been able to find a value for this, well before the invention of calculators. Here I've got a slight problem or, actually, a very big problem! The theory behind this forms one of the harder parts of the A-level maths course, so unless you've got a few years to spare . . .
Hoping this helps,
Chris (a former maths teacher).
So what about powers like two thirds? Well, 8 to the power of two thirds is a calculation which is worked out in two stages. The first part is to calculate 8 to the power of 2 (which equals 64). The second part is to work out 64 to the power of one third (which gives the answer, 4).
Let's try one more like that but where the numbers don't work out quite so easily: If we look at 5 to the power of 1.5, we can re-write 1.5 as a fraction, so we've got 5 to the power of (3 over 2). It's a two-stage calculation, like before, so the first bit is to work out 5 to the power of 3. This gives us 125. The second part is to work out 125 to the power of one half (i.e. the square root of 125). My calculator gives this as 11.180 (to 5 significant figures)
Now, back to your question: 5 to the power of 0.68 means 5 to the power of 68 hundredths. So, the first stage is to work out 5 to the power 68. That's far too big a number to type out so I'll just call the answer ' x '. So we've now got this massive number but we've still got to do the second part of the calculation. i.e. we've got to work out the hundredth root of that great big number, x. At this point, I have to cheat a bit and reach for my calculator to find an answer of 2.9874 (to 5 significant figures).
Hopefully, I've answered the first part of your question, regarding the meaning of 5 to the power of 0.68. What I haven't done is explain how mathematicians have been able to find a value for this, well before the invention of calculators. Here I've got a slight problem or, actually, a very big problem! The theory behind this forms one of the harder parts of the A-level maths course, so unless you've got a few years to spare . . .
Hoping this helps,
Chris (a former maths teacher).
Me again!
I've just re-read your last post, QM, and realised that you want to know which calculator keys Kempie & I used to get our answer of about 2.897. You also say, however, that you have a 'simple' calculator. If this is the case you won't have the right key on your calculator. You need a scientific calculator. The key sequence is very simple:
5 > 'funny key' > 0.68 > =
That 'funny key' is marked in different ways on different scientific calculators. It could be 'Exp' or 'x to the power y' or, on my calculator, 'x to the power square'. (Square, here, just means a picture of a liitle white square).
Chris
I've just re-read your last post, QM, and realised that you want to know which calculator keys Kempie & I used to get our answer of about 2.897. You also say, however, that you have a 'simple' calculator. If this is the case you won't have the right key on your calculator. You need a scientific calculator. The key sequence is very simple:
5 > 'funny key' > 0.68 > =
That 'funny key' is marked in different ways on different scientific calculators. It could be 'Exp' or 'x to the power y' or, on my calculator, 'x to the power square'. (Square, here, just means a picture of a liitle white square).
Chris
Greetings QM! You are lucky I'm here, because I don't normally visit the "How it Works" section (you should have tried the "Science" section (There are too many sections, Mr Editor)). I might as well give an answer, even if it duplicates what other people have said.
Reciprocals? Roots? Logarithms? Antilogs? Converging series? I have literally no idea what you are talking about. I'm not so much a mathophobe as a mathematical cretin.
Reciprocal = "one over something"
reciprocal of 2 = 1/2 = 0.5 = a half
reciprocal of 3 = 1/3 = 0.333333 = a third
reciprocal of 10 = 1/10 = 0.1 = a tenth
similarly
reciprocal of 0.5 = 2
reciprocal of 0.25 = 4
reciprocal of 0.2 = 5
reciprocal of 0.001 = 1000
Root = a smaller number that you need to multiply by itself to get to a bigger number, e.g.
3 x 3 = 9 therefore 3 is the "square root" of 9
4 x 4 = 16 therefore 4 is the "square root" of 16
100 x 100 = 10,000 so 100 is the "square root" of 10,000
2 x 2 x 2 = 8, so 2 is the "cube root" of 8
5 x 5 x 5 = 125, so 5 is the "cube root" of 125
2 x 2 x 2 x 2 = 16, so 2 is the 4th root of 16
10 is the fifth root of 100,000
20 is the tenth root of 10,240,000,000,000
But a root can also be larger than the thing which it multiplies to get to, if the number is smaller than one you start with:
0.5 x 0.5 = 0.25 (1/2 x 1/2 = 1/4)
therefore 0.5 (1/2) is the square root of 0.25 (1/4)
0.2 x 0.2 x 0.2 = 0.008 (1/5 x 1/5 x 1/5 = 1/125)
therefore 0.2 (1/5) is the cube root of 0.008 (1/125)
0.1 x 0.1 x 0.1 x 0.1 = 0.0001 (1/10 x 1/10 x 1/10 x 1/10 = 1/10,000)
therefore 0.1 (1/10) is the fourth root of 0.0001
A logarithm is a smaller number that makes it easier to multiply large numbers, especially before calculators existed. e.g.
100 = 10 x 10, so the "logarithm of 100" is 2.
1000 = 10 x 10 x 10, so the "logarithm of 1000" is 3.
but
100 x 1000 = 100,000
and
2 + 3 = 5
If you do not have a calculator to do it, you can work it out with logarithms instead. So
100 x 1000 = ?????
log of 100 = 2
log of 1000 = 3
2 + 3 = 5
anti-log of 5 = 100,000
So to work out 100 x 1000, you look up the logs in your log table (2 and 3), add them up (5) and then look up 5 in your anti-log table. It will tell you that 5 is the log of 100,000. This thing about logs and anti-logs is based on the idea that it is easier for children to add small numbers than multiply large numbers.
But the ones I have mentioned are all logarithms to the base of 10. You can have logrithms to any base. So
logarithm (to the base 2) of 4 = 2
logarithm (to the base 2) of 8 = 3
logarithm (to the base 2) of 16 = 4
Converging series? I could explain, but it would bore you to death and is not very relevant to the question you asked.
a to the power of b
is written as
a^b
in printed stuff (not on the internet) it is written as an a with a little b up in the air. So
4^2 = 4 squared = 4 x 4
2^3 = 2 cubed = 2 x 2 x 2
5^6 = "5 to the power of 6" = 5 x 5 x 5 x 5 x 5 x 5
but (usefully)
a to the power of b
multiplied by
a to the power of c
equals
a to the power of (b+c)
e.g. 5^7 x 5^15 = 5^22
(this is also what logarithms are about)
Remember what I said about roots of small numbers? The root of a number which is smaller than one is larger than the number. Similarly, a power which is smaller than one also produces a smaller number. So:
10^2 x 10^3 = 10^5 (100 x 1000 = 100,000)
10^1 x 10^1 = 10^2 (10 x 10 = 100)
but also
10^0.5 x 10^0.5 = 10^1
("ten to the power of a half"
times "ten to the power of a half"
equals "ten to the power of one")
or
10^0.3333 x 10^0.3333 x 10^0.3333 = 10^1
("ten to the power of a third"
times "ten to the powert of a third"
times "ten to the power of a third"
equals "ten to the power of one")
From the last two equations, you will / should be able to realise / notice that
10^0.5 (ten to the power of a half)
is
the square root of 10
and
10^0.3333 (ten to the power of a third)
is
the cube root of 10
so
10 ^ 0.1 (ten to the power of a tenth)
is
the tenth root of 10
and also
10 ^ 0.01 (ten to the power of one-hundredth)
is
the 100th root of 10
In other words,
the square root of something is the number that you have to multiply by itself twice in order to get the number you are looking for, and
the cube root of something is the number that you have to multiply by itself three times in order to get the number you are looking for, and
the tenth root (bla bla bla) ten times, and
the hundredth root (bla bla bla) hundred times, and
the nth rooth (bla bla bla) n times.
But remember that
a to the power of b
multiplied by
a to the power of c
equals
a to the power of (b+c)
?
We can use this to work out your answer. So:
5 to the power of 0.68
= 5 ^ 0.68
= (5 ^ 0.6) x (5 ^ 0.08)
but
5 ^ 0.6 = (5 ^ 0.1) x (5 ^ 0.1) .... (six times)
and
5 ^ 0.08 = (5 ^ 0.01) x (5 ^ 0.01) .... (eight times)
or
5 ^ 0.68 = (5 ^ 0.01) x (5 ^ 0.01) .... (68 times)
Actually, the hundredth root of 5 is 1.01622
and 5 ^ 0.68 is 2.98744.
because 1.01622 multiplied by itself 68 times is 2.98744.
If you want to use logarithms:
log 5 = 0.69897
log 2.98744 = 0.475299
log 1.01622 = 0.0069897
You will have noticed that
the log of 5 is 100 times the log of (5 ^ 0.01),
and the log of (5 ^ 0.68) is 68 times the log of (5 ^ 0.01).
5 to the power of 0.68
= 5 ^ 0.68
= (5 ^ 0.6) x (5 ^ 0.08)
but
5 ^ 0.6 = (5 ^ 0.1) x (5 ^ 0.1) .... (six times)
and
5 ^ 0.08 = (5 ^ 0.01) x (5 ^ 0.01) .... (eight times)
or
5 ^ 0.68 = (5 ^ 0.01) x (5 ^ 0.01) .... (68 times)
Actually, the hundredth root of 5 is 1.01622
and 5 ^ 0.68 is 2.98744.
because 1.01622 multiplied by itself 68 times is 2.98744.
If you want to use logarithms:
log 5 = 0.69897
log 2.98744 = 0.475299
log 1.01622 = 0.0069897
You will have noticed that
the log of 5 is 100 times the log of (5 ^ 0.01),
and the log of (5 ^ 0.68) is 68 times the log of (5 ^ 0.01).
Good lord, B! I'm definitely going to have to print out all the answers above, so that I can read them at leisure. There's just too much there to grasp directly from a computer screen for me.
As to why I asked, it was because I had twice in recent days seen the ^0.68 figure. I have no interest in fishing whatsoever, but one of them was on an angling website - God alone knows how I got there - and related to assessing the weight of a fish based on length and girth, each of which had a ^0.X attached to it. I wasn't too sure how one could multiply something by itself less than once...as it seemed to me!
My last maths lesson took place in 1956, so I've succeeded perfectly well in living for all but half a century without any mathematical knowledge beyond simple arithmetic. I have not even once used a quadratic equation in that time. And even when I had a log-book 'way back then, I was invariably on the cosine page when everybody else was on the tangent page (or whatever)!!
Thanks to all of you. Now I'll need to lie down for a while before getting down to trying to understand what you've told me. Then I'll try it out on the next halibut the lady brings home from Sainsbury's!
Well, I did exactly as I said yesterday and copied all your responses and read them through a line at a time a couple of times. I sort of understood at the time, though I wouldn't like to be tested on it today!
Given that I've reached my late sixties without the knowledge you all tried to impart, I can probably scrape through my remaining years without it, too.
However, perhaps one of you could expand on what Chris said about the calculator keys involved. As suspected, my own one did not have the 'funny' key, but I did find an online scientific calculator here Using that, I entered 5 and pressed the x-squared key which gave me the square root of 5. Thereafter, Chris said 0.68 had to be entered but, when I did that, nothing happened except that 0.68 appeared on the screen! I tried multiplying the square root of 5 by 0.68 and dividing it by 0.68 - madness, I know - but neither procedure produced the required number.
So, can anyone outline the complete keying sequence - in idiot-level words - for me, by reference to the online calculator mentioned above? Many thanks to whoever comes up with the goods but also to everyone else involved. Three stars all round...though I know Bernardo disapproves of them!
P.S. (if your mind is not boggling too much)
Did you know that you can also have something to the power of a negative number?!? For example:
10 ^ 3 = 1000
10 ^ 2 = 100
10 ^ 1 = 10
10 ^ 0 = 1
10 ^ -1 = 0.1
10 ^ -2 = 0.01
10 ^ -3 = 0.001
etc.
You will notice that
a to the power of minus b [ a ^ -b ]
is the same as
one over (a to the power of b) [ 1 / ( a ^ b ) ]
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