WEBVTT

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In this particular session,
we're going to be looking

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at indices or powers.

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Either name is used. Both mean

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the same. Basically there
a shorthand way of

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writing. Multiplications of the
same number. So here we have 4

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multiplied by itself three
times. So we write that as 4

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to the power three, so it's
three. That is the power or the

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index. That's the index or the
power. We can do this with

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letters, so we might have a
times a times a times a Times A

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and that's a multiplied by
itself five times. So we write

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that as A to the power 5.

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Do we have something like
2 X squared raised to

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the power four, let's say?

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Then that would mean two X
squared multiplied by two X

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squared multiplied by two X
squared multiplied by two X

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squared 1234 of them all
together. So we can do the tools

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together. 2 * 2 * 2 * 2.
That gives us 16 and X squared

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times by X squared is X to the

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power 4. Times by another X
squared is X to the power 6

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times by another. X squared is X
to the power 8.

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OK, we've got a notation. We've
got a way of writing something

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down. Now when mathematicians
have a notation and they got a

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way of writing something down,
they want to be able to use it

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for other purposes. So for
instance, what might A to the

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minus 2 mean?

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We know what A to the power two
would mean, but what about A to

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the minus two? What would that

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mean? What would something like
A to the power half me?

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What might something like A to
the power 0 mean?

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Well, we need some rules to
operate with an out of looking

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at these rules will find what
these particular notations

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actually mean. So let's begin
with our first rule. Supposing

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we have a cubed and we
want to multiply it by A

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squared, what's our result?

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Well, we know what I cubed is.
That means a times a times a

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three times times by A squared.
So that's a times by a on the

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end there, and altogether we've
got five of them A to the power

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5. And that suggests our very
first rule that if we're

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multiplying together expressions
such as these, then we add the

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indices and so if we have A
to the M times by A to

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the N and the result is A
to the N plus N, and that's

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our first rule.

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Let's have a look at our second
rule. Already done something

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like this previously. Supposing
we had A to the power four, and

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we want to raise it all to the
power three, and we know what

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that means. It means A to the
power four times by A to the

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power four times by A to the

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power 4. Now the first rule
tells us that we should add the

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indices together, so that's A to
the power twelve. 4 + 4 + 4.

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But twelve is 3 times by 4, so
that suggests to us that we

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should, perhaps, if we've got A
to the power M.

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Raised to the power N,
then the result we get

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by multiplying those two
together and that.

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Is our second rule.

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Let's now have a look.

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And our third rule.

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For this, let's take A to the

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7th. And let's divide it by
A to the power three or a

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cubed well, A to the 7th means
a multiplied by itself.

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Seven times.
Divided by so let's divide it

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by. A multiplied by itself 3
times and now we can begin to

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cancel some common factors. So
there's a common factor of A and

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again, there's another common
factor of A and again, there's

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another common factor of a. So
on the bottom here we've really

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got. 1 one and one suite. 1 *
1 is one and on the top a Times

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by a times by a Times by AA to
the power 4.

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But Seven takeaway three is A to
the power four, and so that

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gives us our third rule that if
we have A to the power M divided

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by A to the power N, we get
the result A to the power M

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minus N. And so there's
our third rule.

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OK, we got 3 rules. Let's see
what we can do with them.

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Let's have a look
at a cubed divided

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by a cubed.

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When we know the answer to that,
a cubed divided by a cube we're

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dividing something by itself. So
the answer is got to be 1.

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Fine, let's do it using our laws
of indices, our rules, and we

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can use Rule #3 for this that if
we want to do this, we subtract

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the indices. So that's A to the
power 3 - 3, which is A to

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the power 0.

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So what have we done? We've done
the same calculation in two

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different ways. We've done it
correctly in two different ways,

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so the answers that we get, even
if they look different, must be

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the same. And So what we have is
A to the power 0 equals 1.

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Our 4th result. If you like what
does this mean?

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Any fact it means that any
number raised to the power zero

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is one. So if we take two as we
raise it to the power zero, the

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answer is 1.

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If we take a million and
raise it to the power zero,

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the answer is 1.

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If we take something like a half
and raise it to the power zero,

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the answer is again one we take
minus six and raise it to the

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power zero. The answers one.

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If we take zero and raise it to

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the power 0. Well, it's a bit
complicated, so we'll leave that

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one on side for the moment. Just
bear in mind any number apart

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from zero when raised to the
power zero is equal to 1.

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Let's have a look
now at doing a

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division.

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Again. Let's take the example
that we use when we looked at

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law three, except let's turn it
round, let's do the division the

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other way about. A cubed divided
by A to the 7th.

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Well, you can set that out as we
did before, except it will be

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the other way up. So we have a
cubed is a Times by a times by a

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divided by. A multiplied

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by itself. 7.

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Times. Again, we can do
the canceling, canceling out the

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common factors, dividing top and
bottom by the common factors.

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So what do we have? One on
the top 1234 on the bottom A

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to the Power 4?

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We know we've done that right?
Let's use our third law, our

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third rule, and do it by
subtracting the indices.

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Well, three takeaway 7 is minus
four, so we've got A to the

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power minus four. So same
argument applies. We've done the

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calculation. Same calculation in
two different ways.

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We've done it correctly. We've
arrived at two different answers

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there for these two answers.
Have got to be the same. So

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one over 8 to the power four
is a till they minus 4.

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So a minus sign in the index
with the power means one over

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one over A to the power four.
Let's just develop that one a

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little bit. Let's just look at
one or two examples. So for

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instance, 2 to the power minus
two would be one over.

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Two Square, which of course
gives us a quarter.

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5 to the power minus one
is one over 5 to the

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one which is just one
over 5.

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One

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over. Hey.
Is A to the minus one turning

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it round working backwards?

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One over 7 squared would be
one over 49, but what about

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one over 7 to the minus
2 - 2 remember means one

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over 7 squared's. This is one
over one over 7 squared, and

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here we're dividing by a
fraction and to divide by a

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fraction, we know that we.

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Invert and multiply and so 7
times by 7 is 49 times

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by the one leaves us with
49 or just 7 squared.

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So some examples there. This is
probably the one that you need

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to remember and need to work
with most. It's the basic case

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and if you can remember that one
then they nearly all follow from

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that. So that's that one. Let's
now go on.

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And have a look.

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A tower 6 result.

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What do we mean
by A to the

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power 1/2? What's that mean? So
far we've been working with

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integers an with negative

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numbers. What about A to the
power 1/2 well?

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Supposing we had A to the P and
we multiplied it by A to the P,

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the answer we got was just a.

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Just a That's A to the

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power one. And a Times

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by a. Each with a P on A to the
P times by A to the P using our

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rule would be A to the 2P.

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So 2P must be the same as one.
In other words, P is 1/2.

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What do we have some sort of
interpretation for? This two

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numbers, identical that multiply
together to give a? Well, that's

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the square root, isn't it?

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It's a square root. It's like 7
times by 7 equals 49.

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So if we take that one on.

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7 times by 7 is 49. What
we've got then is 49 to the

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half is equal to 7, so ETA
the half is equal to the square

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root of A.

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A to the third would be equal
to the cube root of A.

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So if we were asked what
is 16 to the quarter?

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What we're asking is what
number, when multiplied by

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itself four times, gives us 16.
A fairly obvious choice for that

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is 2. What

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about 81? To
the half? Well, that's the

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square root of 81, and the
square root of 81 is 9.

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What about 243
and will make

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that to the

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5th? What number, when
multiplied by itself five times,

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will give us 240 three? Well, as
we look at this, we can see it

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divides by three and three's
into that give us 81, so we know

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this is 3 times by 81.

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81 we know.

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Is 9 times by 9.

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And each of
those nines is

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3 times by

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three. Which means the number
that we want is just three.

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Noticing doing this, how
important it is to be able to

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recognize what numbers are made
up of to be able to recognize

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that 16 is 2 to the power,
four that it's also 4 to the

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power to the nine is 3 squared.
That 81 is 9 squared and also

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3 to the power 4.

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You'll find calculations much,
much easier if you can recognize

00:16:40.390 --> 00:16:47.067
in numbers their composition as
powers of simple numbers such as

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two and three, four, and five.

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Once you've got those firmly
fixed in your mind.

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This sort of calculation becomes

00:16:58.325 --> 00:17:01.146
relatively straightforward. 1

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final result. If
we now know what A to the half

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isn't A to the third and eight
to the quarter, what do we mean

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if we take A to the 3/4?

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Well, the quarters alright.
Let's split that up. That means

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A to the power 1/4.

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Cubed and what we're doing is
we're using this result that A

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to the M raised to the power N
is A to the MN. In other words,

00:17:34.966 --> 00:17:38.893
we're using our 2nd result to be
able to do that.

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So let's have a look at an
example using this. Supposing we

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take 60, we say we want 16
to the power 3/4.

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Then that 16 to the power

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quarter. To be cubed, so we
look at this first 16 to the

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Power 1/4 is 2. That's the
number when multiplied by itself

00:18:09.577 --> 00:18:15.176
four times will give us 16
raised to the power 3.

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And that of course is 8 because
that means 2 * 2 * 2.

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But we can think of this another
way, because Eminem can be

00:18:25.104 --> 00:18:29.988
interchanged. Currently we could
write this as A to the power N

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raised to the power M.

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That would be just the same
result, so we can think of this

00:18:39.359 --> 00:18:44.989
in a slightly different way.
Let's take a different example.

00:18:44.989 --> 00:18:46.678
Let's take 8.

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To the power 2/3.

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Now, the way that we had thought
about that was to do 8 to the

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power third. Then square it.

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And we know that 8 to the
power 1:30 is 2 and 2

00:19:05.665 --> 00:19:08.040
squared is there for four.

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We don't have to think of it
like that. We can think of it

00:19:15.472 --> 00:19:20.455
as 8 squared raised to the
Power, 1/3 eight squared we

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know is 64 and now we have to
take it to the power 1/3. We

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need the cube root. We need a
number that when multiplied

00:19:32.233 --> 00:19:37.669
by itself three times gives
us 64 and that number is 4.

00:19:38.750 --> 00:19:41.410
These two are equivalent.

00:19:41.930 --> 00:19:47.690
So the different interpretations
that we've got are the same.

00:19:47.690 --> 00:19:52.060
Writing it down algebraically,
what we're saying is that we

00:19:52.060 --> 00:19:58.941
have. A letter of variable to
the power P over Q. Then we can

00:19:58.941 --> 00:20:05.346
write that in one of two ways.
We can write it as either A to

00:20:05.346 --> 00:20:06.627
the power P.

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Find the cube root of it, which
might be written as A to the

00:20:14.308 --> 00:20:21.476
power P. Find the cube root, or
it might be written as a. Let's

00:20:21.476 --> 00:20:27.620
take the cube root and raise it
all to the power P.

00:20:28.160 --> 00:20:34.656
So we might have a find the cube
root of it and raise it all to

00:20:34.656 --> 00:20:35.874
the power P.

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Either result is exactly the

00:20:39.460 --> 00:20:46.286
same. Now, to conclude, let's
just have a look at some very

00:20:46.286 --> 00:20:48.230
basic simple calculations using

00:20:48.230 --> 00:20:54.536
indices. 2X to the minus 1/4 and
our objective here is to write

00:20:54.536 --> 00:20:59.732
it with a positive index. Well,
the two doesn't have an index

00:20:59.732 --> 00:21:04.928
attached to it at all, so
nothing happens to the two just

00:21:04.928 --> 00:21:10.990
stays it is X to the minus 1/4.
The minus sign means one over

00:21:10.990 --> 00:21:16.619
right it in the denominator, so
we write it down there. So we

00:21:16.619 --> 00:21:19.650
have two over X to the power

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quarter. 4X to the minus 2A
cubed. Well, nothing wrong with

00:21:25.170 --> 00:21:31.194
the four and nothing wrong with
the A cubed at perfectly normal

00:21:31.194 --> 00:21:34.206
so they stay as they are.

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X to the minus 2.

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A minus in the index and so
that means one over, so it's 4A

00:21:44.706 --> 00:21:46.842
cubed over X squared.

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One over 4A to the
power minus 2.

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Well, we met this one before.

00:21:59.340 --> 00:22:06.132
If you remember, we actually
looked at one over 7 to the

00:22:06.132 --> 00:22:13.435
minus 2. And that was
one over one over 7 squared

00:22:13.435 --> 00:22:20.455
and because we were dividing by
a fraction, we said we had

00:22:20.455 --> 00:22:27.475
to invert and multiply. And so
we ended up with 7 squared.

00:22:28.440 --> 00:22:33.060
This is no different. As a
letter there instead of a

00:22:33.060 --> 00:22:37.260
number, the result is going to
be exactly the same.

00:22:37.830 --> 00:22:44.942
So the four stays where it is
one over 4 and this becomes a

00:22:44.942 --> 00:22:50.530
squared and to write it in a
more simple, tidier fashion,

00:22:50.530 --> 00:22:53.070
it's A squared over 4.

00:22:54.130 --> 00:23:01.800
A to the minus 1/3
times by two A to

00:23:01.800 --> 00:23:04.868
the minus 1/2 equals.

00:23:05.480 --> 00:23:13.144
Two, we can just write down A to
the minus 1/3 times by 8 in the

00:23:13.144 --> 00:23:18.892
minus 1/2 hour. First job is to
add those indices together, so

00:23:18.892 --> 00:23:21.287
minus 1/3 plus minus 1/2.

00:23:21.930 --> 00:23:29.602
He's going to give us 2A to
the power now. A third and a

00:23:29.602 --> 00:23:36.726
half is 5 six. So with the
minus signs, that is minus five

00:23:36.726 --> 00:23:43.302
sixths, and so that's two over A
to the power five sixths.

00:23:44.270 --> 00:23:50.774
2A to the
minus 2 divided

00:23:50.774 --> 00:23:57.278
by A to
the minus three

00:23:57.278 --> 00:24:04.794
over 2. Well, that's
2A to the minus 2 divided

00:24:04.794 --> 00:24:08.352
by A to the minus three

00:24:08.352 --> 00:24:14.584
over 2. Let's remember what our
rules said that if we're

00:24:14.584 --> 00:24:18.958
dividing things like this, we
actually subtract the indices.

00:24:18.958 --> 00:24:25.762
So this is 2A to the minus 2
minus minus three over two. Of

00:24:25.762 --> 00:24:31.108
course, that means effectively
we've got to add on to three

00:24:31.108 --> 00:24:38.884
over 2, so we get 2A to the
minus 2 + 3 over 2 inches minus

00:24:38.884 --> 00:24:40.828
1/2, so that's 2A.

00:24:40.840 --> 00:24:44.240
Over A to the half.

00:24:44.790 --> 00:24:52.326
Now then we take
something that looks a

00:24:52.326 --> 00:24:55.152
little bit complicated.

00:24:55.820 --> 00:25:03.370
So what do we got
here? We've got the cube

00:25:03.370 --> 00:25:10.920
root of A squared times
by the square root of

00:25:10.920 --> 00:25:17.560
a cube. First of all, let's
write these as indices. This is

00:25:17.560 --> 00:25:23.397
the cube root of A squared. So
that means a squared and take

00:25:23.397 --> 00:25:27.887
the cube root. You know it's A
to the 2/3.

00:25:28.520 --> 00:25:35.540
Times by a now this is
a cubed. Take the square root

00:25:35.540 --> 00:25:39.050
so that's a cubed. Take the

00:25:39.050 --> 00:25:46.396
square root. And we can see
now what we need to do is add

00:25:46.396 --> 00:25:52.350
together these two indices. So
that's 2/3 + 3 over 2 and adding

00:25:52.350 --> 00:25:57.846
fractions together. That will
give us A to the power 13 over

00:25:57.846 --> 00:26:03.800
6 awkward number will just leave
it like that for now. Now let's

00:26:03.800 --> 00:26:09.296
have a look at some calculations
using numbers. This time 16 to

00:26:09.296 --> 00:26:13.876
the Power 3/4. Now again, we've
already done this one.

00:26:14.810 --> 00:26:18.560
That 16 to the power 1/4.

00:26:19.120 --> 00:26:23.719
Cubed what number, when
multiplied by itself four times,

00:26:23.719 --> 00:26:30.362
gives us 16? That's two raised
to the power. Three gives us 8

00:26:30.362 --> 00:26:31.895
for our answer.

00:26:32.680 --> 00:26:39.583
4 to the power minus five over
2. Let's deal with the minus

00:26:39.583 --> 00:26:45.424
sign first. Minus sign means one
over. Put it until the

00:26:45.424 --> 00:26:48.610
denominator, 4 to the power 5

00:26:48.610 --> 00:26:55.080
over 2. One over 4 to
the half to the power five

00:26:55.080 --> 00:27:00.924
notice each time I'm doing the
root bit first, the 4th root

00:27:00.924 --> 00:27:05.794
here, the square root here.
That's because by doing the

00:27:05.794 --> 00:27:11.151
route, I get the number smaller.
I can handle the arithmetic

00:27:11.151 --> 00:27:18.456
easier, so this is now one over
the square root of 4 is 2 raised

00:27:18.456 --> 00:27:20.404
to the power 5.

00:27:20.440 --> 00:27:24.388
And that's one over 32 again.
Notice I wrote it straight down

00:27:24.388 --> 00:27:28.336
without seeming to work it out.
That's because I know what it

00:27:28.336 --> 00:27:32.284
is. And again, it's another one
of these relationships. 2 to the

00:27:32.284 --> 00:27:36.890
power, three is 8 two to the
power, five is 32 to the power,

00:27:36.890 --> 00:27:40.838
six is 64 that really, if you
can learn and get become

00:27:40.838 --> 00:27:44.128
familiar with you, find this
kind of calculation much easier.

00:27:45.610 --> 00:27:53.420
Here's another one. 125 seems
a very big number to

00:27:53.420 --> 00:28:00.546
the 2/3. Equals 125
to the power of

00:28:00.546 --> 00:28:03.054
3rd all squared.

00:28:03.750 --> 00:28:09.067
Now, even if we've never met 125
before, in these terms, one of

00:28:09.067 --> 00:28:15.611
things we ought to be aware of
is it ends in a 5, so it must

00:28:15.611 --> 00:28:20.519
divide by 5. So it's a fair
guess that 5 multiplied by

00:28:20.519 --> 00:28:25.836
itself three times would give us
125, and indeed it does so to

00:28:25.836 --> 00:28:30.744
the third power, 125 to the
third power is 5. The number

00:28:30.744 --> 00:28:31.971
multiplied by itself.

00:28:32.640 --> 00:28:39.195
Three times that gives us 125 is
5 and now we just want to square

00:28:39.195 --> 00:28:41.817
that and that obviously gives us

00:28:41.817 --> 00:28:49.580
25. Take 8 - 2/3
again. Let's deal with the minus

00:28:49.580 --> 00:28:55.295
sign. That is one over 8
to the 2/3.

00:28:55.860 --> 00:29:01.349
Again, let's deal with the
third bit, the root bit first.

00:29:01.349 --> 00:29:05.840
What number, when multiplied
by itself three times, would

00:29:05.840 --> 00:29:12.826
give us eight that must be 2
Ann. We need to square it and

00:29:12.826 --> 00:29:14.822
so that becomes 1/4.

00:29:18.490 --> 00:29:24.774
We take
one over

00:29:24.774 --> 00:29:31.058
25 to
the minus

00:29:31.058 --> 00:29:38.707
2. Well, remember we
saw what this happened with the

00:29:38.707 --> 00:29:46.063
one over 7 to the minus
two this became 25 squared and

00:29:46.063 --> 00:29:48.515
that is simply 625.

00:29:50.190 --> 00:29:53.610
243 to

00:29:53.610 --> 00:30:01.550
the 3/5.
Let's do the 5th bit 1st

00:30:01.550 --> 00:30:07.900
and then the QB. Well 243,
that's three multiplied by

00:30:07.900 --> 00:30:15.520
itself five times. So that is
a three race to the power

00:30:15.520 --> 00:30:18.060
3, and that's 27.

00:30:19.390 --> 00:30:26.644
One last example,
let's take 81

00:30:26.644 --> 00:30:30.271
and to the

00:30:30.271 --> 00:30:37.174
minus 3/4.
OK, minus sign means

00:30:37.174 --> 00:30:43.992
one over. Where dividing
by a fraction to

00:30:43.992 --> 00:30:50.728
divide by a fraction,
we invert and multiply,

00:30:50.728 --> 00:30:57.464
so this becomes 16
over 81 raised to

00:30:57.464 --> 00:30:59.990
the power 3/4.

00:31:00.050 --> 00:31:07.050
Equals 16 over 81.
First of all to

00:31:07.050 --> 00:31:10.550
the quarter and then.

00:31:10.560 --> 00:31:15.824
Raised to the power three to the
quarter, this is 2 to the power

00:31:15.824 --> 00:31:21.088
four on top 16 is 2 to the power
four. So that's a 2.

00:31:21.860 --> 00:31:28.548
And 81 is 3 to the power four,
so that's a 3 and we need to

00:31:28.548 --> 00:31:32.310
cube that and that gives us
eight over 27th.

00:31:33.130 --> 00:31:37.243
So working with these indices
shouldn't really be too

00:31:37.243 --> 00:31:42.270
difficult. You need to remember
how numbers are made up. You

00:31:42.270 --> 00:31:48.211
need to have in your mind's eye
that 243 is 3 multiplied by

00:31:48.211 --> 00:31:54.152
itself. 5 * 243 is 3 to the
power five, and similarly with

00:31:54.152 --> 00:31:56.437
numbers like 30, two, 64128.

00:31:57.750 --> 00:32:02.958
If you can carry those around in
your head and you will do with

00:32:02.958 --> 00:32:07.422
practice, you get to know them
quite well. Then this kind of

00:32:07.422 --> 00:32:11.142
calculation should become very
easy to remember. Deal with the

00:32:11.142 --> 00:32:15.606
minus sign first. By and large.
Get that to be positive, then

00:32:15.606 --> 00:32:19.698
deal with the root sign that
gets the number. Work down.

00:32:19.698 --> 00:32:20.814
Keeps it small.

00:32:21.740 --> 00:32:24.435
They can be tricky, but we
hope we've shown that they

00:32:24.435 --> 00:32:25.170
can be mastered.