www.mathcentre.ac.uk/.../powers.mp4

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 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 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.