In this particular session, we're going to be looking
at indices or powers.
Either name is used. Both mean
the same. Basically there a shorthand way of
writing. Multiplications of the same number. So here we have 4
multiplied by itself three times. So we write that as 4
to the power three, so it's three. That is the power or the
index. That's the index or the power. We can do this with
letters, so we might have a times a times a times a Times A
and that's a multiplied by itself five times. So we write
that as A to the power 5.
Do we have something like 2 X squared raised to
the power four, let's say?
Then that would mean two X squared multiplied by two X
squared multiplied by two X
squared 1234 of them all together. So we can do the tools
together. 2 * 2 * 2 * 2. That gives us 16 and X squared
times by X squared is X to the
power 4. Times by another X squared is X to the power 6
times by another. X squared is X to the power 8.
OK, we've got a notation. We've got a way of writing something
down. Now when mathematicians have a notation and they got a
way of writing something down, they want to be able to use it
for other purposes. So for instance, what might A to the
minus 2 mean?
We know what A to the power two would mean, but what about A to
the minus two? What would that
mean? What would something like A to the power half me?
What might something like A to the power 0 mean?
Well, we need some rules to operate with an out of looking
at these rules will find what these particular notations
actually mean. So let's begin with our first rule. Supposing
we have a cubed and we want to multiply it by A
squared, what's our result?
Well, we know what I cubed is. That means a times a times a
three times times by A squared. So that's a times by a on the
end there, and altogether we've got five of them A to the power
5. And that suggests our very first rule that if we're
multiplying together expressions such as these, then we add the
indices and so if we have A to the M times by A to
the N and the result is A to the N plus N, and that's
our first rule.
Let's have a look at our second rule. Already done something
like this previously. Supposing we had A to the power four, and
we want to raise it all to the power three, and we know what
that means. It means A to the power four times by A to the
power 4. Now the first rule tells us that we should add the
indices together, so that's A to the power twelve. 4 + 4 + 4.
But twelve is 3 times by 4, so that suggests to us that we
should, perhaps, if we've got A to the power M.
Raised to the power N, then the result we get
by multiplying those two together and that.
Is our second rule.
Let's now have a look.
And our third rule.
For this, let's take A to the
7th. And let's divide it by A to the power three or a
cubed well, A to the 7th means a multiplied by itself.
Seven times. Divided by so let's divide it
by. A multiplied by itself 3 times and now we can begin to
cancel some common factors. So there's a common factor of A and
again, there's another common factor of A and again, there's
another common factor of a. So on the bottom here we've really
got. 1 one and one suite. 1 * 1 is one and on the top a Times
by a times by a Times by AA to the power 4.
But Seven takeaway three is A to the power four, and so that
gives us our third rule that if we have A to the power M divided
by A to the power N, we get the result A to the power M
minus N. And so there's our third rule.
OK, we got 3 rules. Let's see what we can do with them.
Let's have a look at a cubed divided
by a cubed.
When we know the answer to that, a cubed divided by a cube we're
dividing something by itself. So the answer is got to be 1.
Fine, let's do it using our laws of indices, our rules, and we
can use Rule #3 for this that if we want to do this, we subtract
the indices. So that's A to the power 3 - 3, which is A to
the power 0.
So what have we done? We've done the same calculation in two
different ways. We've done it correctly in two different ways,
so the answers that we get, even if they look different, must be
the same. And So what we have is A to the power 0 equals 1.
Our 4th result. If you like what does this mean?
Any fact it means that any number raised to the power zero
is one. So if we take two as we raise it to the power zero, the
answer is 1.
If we take a million and raise it to the power zero,
the answer is 1.
If we take something like a half and raise it to the power zero,
the answer is again one we take minus six and raise it to the
power zero. The answers one.
If we take zero and raise it to
the power 0. Well, it's a bit complicated, so we'll leave that
one on side for the moment. Just bear in mind any number apart
from zero when raised to the power zero is equal to 1.
Let's have a look now at doing a
division.
Again. Let's take the example that we use when we looked at
law three, except let's turn it round, let's do the division the
other way about. A cubed divided by A to the 7th.
Well, you can set that out as we did before, except it will be
the other way up. So we have a cubed is a Times by a times by a
divided by. A multiplied
by itself. 7.
Times. Again, we can do the canceling, canceling out the
common factors, dividing top and bottom by the common factors.
So what do we have? One on the top 1234 on the bottom A
to the Power 4?
We know we've done that right? Let's use our third law, our
third rule, and do it by subtracting the indices.
Well, three takeaway 7 is minus four, so we've got A to the
power minus four. So same argument applies. We've done the
calculation. Same calculation in two different ways.
We've done it correctly. We've arrived at two different answers
there for these two answers. Have got to be the same. So
one over 8 to the power four is a till they minus 4.
So a minus sign in the index with the power means one over
one over A to the power four. Let's just develop that one a
little bit. Let's just look at one or two examples. So for
instance, 2 to the power minus two would be one over.
Two Square, which of course gives us a quarter.
5 to the power minus one is one over 5 to the
one which is just one over 5.
One
over. Hey. Is A to the minus one turning
it round working backwards?
One over 7 squared would be one over 49, but what about
one over 7 to the minus 2 - 2 remember means one
over 7 squared's. This is one over one over 7 squared, and
here we're dividing by a fraction and to divide by a
fraction, we know that we.
Invert and multiply and so 7 times by 7 is 49 times
by the one leaves us with 49 or just 7 squared.
So some examples there. This is probably the one that you need
to remember and need to work with most. It's the basic case
and if you can remember that one then they nearly all follow from
that. So that's that one. Let's now go on.
And have a look.
A tower 6 result.
What do we mean by A to the
power 1/2? What's that mean? So far we've been working with
integers an with negative
numbers. What about A to the power 1/2 well?
Supposing we had A to the P and we multiplied it by A to the P,
the answer we got was just a.
Just a That's A to the
power one. And a Times
by a. Each with a P on A to the P times by A to the P using our
rule would be A to the 2P.
So 2P must be the same as one. In other words, P is 1/2.
What do we have some sort of interpretation for? This two
numbers, identical that multiply together to give a? Well, that's
the square root, isn't it?
It's a square root. It's like 7 times by 7 equals 49.
So if we take that one on.
7 times by 7 is 49. What we've got then is 49 to the
half is equal to 7, so ETA the half is equal to the square
root of A.
A to the third would be equal to the cube root of A.
So if we were asked what is 16 to the quarter?
What we're asking is what number, when multiplied by
itself four times, gives us 16. A fairly obvious choice for that
is 2. What
about 81? To the half? Well, that's the
square root of 81, and the square root of 81 is 9.
What about 243 and will make
that to the
5th? What number, when multiplied by itself five times,
will give us 240 three? Well, as we look at this, we can see it
divides by three and three's into that give us 81, so we know
this is 3 times by 81.
81 we know.
Is 9 times by 9.
And each of those nines is
3 times by
three. Which means the number that we want is just three.
Noticing doing this, how important it is to be able to
recognize what numbers are made up of to be able to recognize
that 16 is 2 to the power, four that it's also 4 to the
power to the nine is 3 squared. That 81 is 9 squared and also
3 to the power 4.
You'll find calculations much, much easier if you can recognize
in numbers their composition as powers of simple numbers such as
two and three, four, and five.
Once you've got those firmly fixed in your mind.
This sort of calculation becomes
relatively straightforward. 1
final result. If we now know what A to the half
isn't A to the third and eight to the quarter, what do we mean
if we take A to the 3/4?
Well, the quarters alright. Let's split that up. That means
A to the power 1/4.
Cubed and what we're doing is we're using this result that A
to the M raised to the power N is A to the MN. In other words,
we're using our 2nd result to be able to do that.
So let's have a look at an example using this. Supposing we
take 60, we say we want 16 to the power 3/4.
Then that 16 to the power
quarter. To be cubed, so we look at this first 16 to the
Power 1/4 is 2. That's the number when multiplied by itself
four times will give us 16 raised to the power 3.
And that of course is 8 because that means 2 * 2 * 2.
But we can think of this another way, because Eminem can be
interchanged. Currently we could write this as A to the power N
raised to the power M.
That would be just the same result, so we can think of this
in a slightly different way. Let's take a different example.
Let's take 8.
To the power 2/3.
Now, the way that we had thought about that was to do 8 to the
power third. Then square it.
And we know that 8 to the power 1:30 is 2 and 2
squared is there for four.
We don't have to think of it like that. We can think of it
as 8 squared raised to the Power, 1/3 eight squared we
know is 64 and now we have to take it to the power 1/3. We
need the cube root. We need a number that when multiplied
by itself three times gives us 64 and that number is 4.
These two are equivalent.
So the different interpretations that we've got are the same.
Writing it down algebraically, what we're saying is that we
have. A letter of variable to the power P over Q. Then we can
write that in one of two ways. We can write it as either A to
the power P.
Find the cube root of it, which might be written as A to the
power P. Find the cube root, or it might be written as a. Let's
take the cube root and raise it all to the power P.
So we might have a find the cube root of it and raise it all to
the power P.
Either result is exactly the
same. Now, to conclude, let's just have a look at some very
basic simple calculations using
indices. 2X to the minus 1/4 and our objective here is to write
it with a positive index. Well, the two doesn't have an index
attached to it at all, so nothing happens to the two just
stays it is X to the minus 1/4. The minus sign means one over
right it in the denominator, so we write it down there. So we
have two over X to the power
quarter. 4X to the minus 2A cubed. Well, nothing wrong with
the four and nothing wrong with the A cubed at perfectly normal
so they stay as they are.
X to the minus 2.
A minus in the index and so that means one over, so it's 4A
cubed over X squared.
One over 4A to the power minus 2.
Well, we met this one before.
If you remember, we actually looked at one over 7 to the
minus 2. And that was one over one over 7 squared
and because we were dividing by a fraction, we said we had
to invert and multiply. And so we ended up with 7 squared.
This is no different. As a letter there instead of a
number, the result is going to be exactly the same.
So the four stays where it is one over 4 and this becomes a
squared and to write it in a more simple, tidier fashion,
it's A squared over 4.
A to the minus 1/3 times by two A to
the minus 1/2 equals.
Two, we can just write down A to the minus 1/3 times by 8 in the
minus 1/2 hour. First job is to add those indices together, so
minus 1/3 plus minus 1/2.
He's going to give us 2A to the power now. A third and a
half is 5 six. So with the minus signs, that is minus five
sixths, and so that's two over A to the power five sixths.
2A to the minus 2 divided
by A to the minus three
over 2. Well, that's 2A to the minus 2 divided
by A to the minus three
over 2. Let's remember what our rules said that if we're
dividing things like this, we actually subtract the indices.
So this is 2A to the minus 2 minus minus three over two. Of
course, that means effectively we've got to add on to three
over 2, so we get 2A to the minus 2 + 3 over 2 inches minus
1/2, so that's 2A.
Over A to the half.
Now then we take something that looks a
little bit complicated.
So what do we got here? We've got the cube
root of A squared times by the square root of
a cube. First of all, let's write these as indices. This is
the cube root of A squared. So that means a squared and take
the cube root. You know it's A to the 2/3.
Times by a now this is a cubed. Take the square root
so that's a cubed. Take the
square root. And we can see now what we need to do is add
together these two indices. So that's 2/3 + 3 over 2 and adding
fractions together. That will give us A to the power 13 over
6 awkward number will just leave it like that for now. Now let's
have a look at some calculations using numbers. This time 16 to
the Power 3/4. Now again, we've already done this one.
That 16 to the power 1/4.
Cubed what number, when multiplied by itself four times,
gives us 16? That's two raised to the power. Three gives us 8
for our answer.
4 to the power minus five over 2. Let's deal with the minus
sign first. Minus sign means one over. Put it until the
denominator, 4 to the power 5
over 2. One over 4 to the half to the power five
notice each time I'm doing the root bit first, the 4th root
here, the square root here. That's because by doing the
route, I get the number smaller. I can handle the arithmetic
easier, so this is now one over the square root of 4 is 2 raised
to the power 5.
And that's one over 32 again. Notice I wrote it straight down
without seeming to work it out. That's because I know what it
is. And again, it's another one of these relationships. 2 to the
power, three is 8 two to the power, five is 32 to the power,
six is 64 that really, if you can learn and get become
familiar with you, find this kind of calculation much easier.
Here's another one. 125 seems a very big number to
the 2/3. Equals 125 to the power of
3rd all squared.
Now, even if we've never met 125 before, in these terms, one of
things we ought to be aware of is it ends in a 5, so it must
divide by 5. So it's a fair guess that 5 multiplied by
itself three times would give us 125, and indeed it does so to
the third power, 125 to the third power is 5. The number
multiplied by itself.
Three times that gives us 125 is 5 and now we just want to square
that and that obviously gives us
25. Take 8 - 2/3 again. Let's deal with the minus
sign. That is one over 8 to the 2/3.
Again, let's deal with the third bit, the root bit first.
What number, when multiplied by itself three times, would
give us eight that must be 2 Ann. We need to square it and
so that becomes 1/4.
We take one over
25 to the minus
2. Well, remember we saw what this happened with the
one over 7 to the minus two this became 25 squared and
that is simply 625.
243 to
the 3/5. Let's do the 5th bit 1st
and then the QB. Well 243, that's three multiplied by
itself five times. So that is a three race to the power
3, and that's 27.
One last example, let's take 81
and to the
minus 3/4. OK, minus sign means
one over. Where dividing by a fraction to
divide by a fraction, we invert and multiply,
so this becomes 16 over 81 raised to
the power 3/4.
Equals 16 over 81. First of all to
the quarter and then.
Raised to the power three to the quarter, this is 2 to the power
four on top 16 is 2 to the power four. So that's a 2.
And 81 is 3 to the power four, so that's a 3 and we need to
cube that and that gives us eight over 27th.
So working with these indices shouldn't really be too
difficult. You need to remember how numbers are made up. You
need to have in your mind's eye that 243 is 3 multiplied by
itself. 5 * 243 is 3 to the power five, and similarly with
numbers like 30, two, 64128.
If you can carry those around in your head and you will do with
practice, you get to know them quite well. Then this kind of
calculation should become very easy to remember. Deal with the
minus sign first. By and large. Get that to be positive, then
deal with the root sign that gets the number. Work down.
Keeps it small.
They can be tricky, but we hope we've shown that they
can be mastered.