www.mathcentre.ac.uk/.../Surds.mp4

Today we're going to look at
numbers, written his powers, and

we're going to go through some
calculations involving them, and

in particular, we're going to
look at square roots.

And the square roots of numbers,
which give us an irrational

answer. That is.

Numbers which can be written as
whole numbers are fractions. Now

these types of square roots are

called Surds. But more of that

later. What I want to do first
is a little bit of revision.

And we'll start with something
that we know.

2 cubed Is written as
2 * 2 * 2.

It's a neat way of writing this

repeated multiplication. And we
say that three is the power or

index. And two is rears to that
power or index.

And the value of 2 cubed
works out to be it because

it's 2 * 2 * 2.

So if we had 4 cubed.

That is 4 * 4

* 4. And we know when that's
worth. Diet is 64.

But what if we have 4 to the

negative 3? What would be its
value and high? Could we put it

down more deeply?

Well, we go back to what we

know. 4 cubed equals 4 * 4
* 4, which is 64.

4 squared equals 4
* 4 and not

equals 16. 14
power one is for.

Is for.

An forwarded the zero. Well, if
we look at our pattern in our

calculations. We are dividing by
for each time.

And the index. Our power
decreases by one. So forward the

zero must be 4 / 4 which

equals 1. But we were interested
in finding out about four to the

negative three, so will continue
our pattern of multiplications.

4 to the negative one will
be the 1 / 4 which

is 1/4. And forward to
the negative two must be 1/4.

Divided by 4, which is 16.

An four to the negative three
must be 116th divided by 4,

which is one over 64.

But If we go back to this
line, we knew that 4 cubed is

64, so we can rewrite one over
64 as one over 4 cubed.

And look at that. Forward
to the negative three is

one over 4 cubed.

So for the negative two is one
over 16 must be one over 4

squared. And four to the
negative, one must equal

for one over 4th power,
one which is one over 4.

So negative powers give you the
reciprocal of the number.

That is one over the number.

So if I had this example.

3 to the negative 2.

We can write that as
one over 3 squared.

And one over 3 squared is the
same as one over 9.

What about this one? 5 to the
negative 3? How can we rewrite

that and what's the value? Well,
the negative 3 means it's one

over. 5 cubed

And that is one over 5 cubed
is 125. So 5 to the negative

three is one over 125.

Not the big thing to note is
that even though you've got a

negative power, that value that
you get is a positive answer.

That's a big mistake by quite a
lot of people. They think when

you've got a negative power,
you'll get a negative answer, so

please don't make that mistake.
And negative power means

reciprocal of the number one
over the power.

Moving on in the examples that
I've just been doing, we've used

positive negative integers for
the powers, but what if we have

fractional indices or fractional

powers? How can we represent
them? What's their value?

Well again will start with what

we know. We know that force of
power one is 4.

But what would be for to the
half? What's it value?

We're looking at this, we

know using. Indices laws,
therefore, to the half times by

4 to the half, must equal 4.

To the indices added together.

And 1/2 + 1/2 is one.

Before to the one is just 4.

So 4 to the half times, 4
1/2 equals 4.

That means that it's something
times by itself equals 4. Well,

we know that it's obvious it's
two 2 * 2 equals 4.

So 4 to the half equals 2.

So. Forza half.

Is equal to 2.

And we write this in a shorthand
notation by doing this for the

half is equal to the square

root. Of four, and that is the
symbol for the square root.

So the half is associated
with the square routing of a

number. So if we had 9 to the
power of 1/2, that's the same

as the square root of 9,
which equals 3.

So in general, if we had a
number a razor, the part half.

Then we can write that as the
square root of A.

We can continue this on by
saying if we wanted the cube

root of a number, say for.

To the third, we can write that
as the cube root of 4.

Like this?

And the fraction associated with
the cube root is 1/3. Similarly,

the 4th root. So if we had
five, the 4th root of 5, you

can write that using this
shorthand notation with a small

for their and then this root
symbol. And the quarter is

associated with the 4th root.

So if we had the NTH root.

Of a number A.

We can write it like
this in shorthand.

So if we have the
cube root of 64.

We can write it like this.

And it equals 64. Raise
the part of 1/3. Now

we know 64 is 4
* 4 * 4.

And that's raised to the power

of 1/3. And the answer is dead
easy. We want to know what

number times by itself three
times gives you 64. Well, it's

obvious because what we've just
written it must be for. So the

cube root of 64 is 4.

A very useful simple fact
leading on from this that.

Is known, but people tend to
forget it is that we know that

forward to the half times by 4
to the half equals 4.

That is the square root of 4
times by the square root of 4 is

4 and we can write that more
simply by saying is the square

root of 4 squared equals 4.

Not, I'll be using this very
simple fact. Little role at the

end of the session, so remember
it. It's any number. The square

root of it squared gives you

that number. Another important
thing to remember about serves

comes from solving this
equation. If we had this very

simple quadratic equation X
squared equal 4.

If we take the square root of
four, we know that we have to

have two routes.

For the solution, because this
is a quadratic equation, so X

equals plus or minus the square
root of 4.

So we know that is
positive two or

negative two. You've
got 2 routes.

So. Roots are not unique.

And remember, we don't
always have to write the

positive side. We can write
positive two as two.

And remember. If you haven't got
a sign in front of the square

root, then it's assumed it's a

positive square root. When it's
not, you have to put the sign in

and make it a negative square

root. But the positive
square roots of the ones

that are are most common.

Now moving on.

What I'd like to do is actually

do some work. On

The square root of negative

numbers. We've done the square
root of positive numbers. What

if we had the square root of

negative 9? Can we evaluate

this? Well, the square root of
negative 9 can be written as

negative 9 raised to the power

of 1/2. And So what we're
looking for are two numbers

multiplied by themselves, which
will give us negative. Now when

we know 3 * 3 is 9.

And negative three times
negative three is 9.

But there's not.

2 numbers multiply by
themselves, which will give us

negative night. So you
cannot evaluate the square

root of a negative number
and get a real answer.

Now what I'd like to do is
continue on using surds in some

calculations. And go over some
common square roots that you

might know. Now we all know
that the square root of 25 is 5

because 5 * 5 is 25.

And if we were given the
square root of 9 over 4,

that's dead easy. It must be
3 over 2 because 3 * 3 is

nine, 2 * 2 is 4, and we
write that as one and a half.

But we've now taken square root
of whole numbers and fractions,

and we've got answers that are
whole numbers and fractions.

But there are other square roots
in which we don't get whole

numbers or fractions. We get
them as non terminating

decimals. Such as Route 2.

Route 2 cannot be evaluated as a
whole number or a fraction.

Never can Route 3.

We can only write them
as approximations. They

are irrational numbers.

An approximation for the square
root of 2 to 3 decimal places

is 1.414. Approximation from
Route 3 to 3 decimal places,

and I put 3 decimal places
just to remind us is 1.732.

Not because. They cannot be
evaluated as whole numbers

and fractions, and these are
irrational numbers.

They are very special and we
call them surged.

Other common starts
would be Route

5. Rich sex, which 7
Route 8 not Route 9 because we

know that's three and Route 10.

Now, why do we need to know

about search? Will search crop
up a lot when we using

Pythagoras Theorem and in
trigonometry and so we need to

know how to manipulate these in

various calculations. And we
need to know how to simplify

them as well.

Now Route 2 cannot be simplified
any further. Now they can Route

3 or Route 5 with six, 3, Seven,
but look at root it. If we take

the square root of it.

It can be replaced by the
product 4 * 2.

Now the square root of product.

Is the product of the square
roots, so it's Square root of 4

times by the square root of 2.

We know the square root of 4 is

2. And this is Route 2. So the
square root of it can be written

more simply as two square root
of two or two Route 2.

In shorthand. What about this
one? What about the square root

of 5 times by the square root

of 15? Now there in the previous
example, the product of the

square roots is equal to the
square root of the products, so

we use that fact too.

Rewrite this expression in

another way. The square root of
5 times by the square root of 15

is the square root of 5 times by

15. And that is the square
root of 75.

But 75 can be rewritten
as 25 times by three.

Note 25 is a square number and
we can then separate that out to

say that is the square root of
25 times by the square root of 3

square root of 25 is 5 times by
Route 3. So the square root of 5

times the square root of 15 is 5
route 3, but could be easier.

But a common mistake is
the people would take an

expression like this.

I served in the form of the
square root of 4 + 9 and say is

equal to the square root of 4
plus the square root of 9.

Now this can't happen
because if we follow this

through that would be 2 +
3, which would equal 5.

This is not correct because five
is the square root of 25.

And here we've got 4 + 9, which
is 13. So the square root of an

addition is not equal to the
square root plus the other

square root. So Please remember

that. Those two rules that we
previously did with Surds, an

expansion of multiplications
only applies to multiplication

and division, not to addition
and Subtraction.

So if we move on.

And we take the square root
of bigger numbers times by.

Square root of a another
big number.

The square root of 400 times by
the square root of 90.

It's useful to remember
square numbers. The

typical square numbers.

Square numbers of the numbers 1
to 10 at one, four 916-2549,

sixty, 481 and 100. You should
be able to remember those

straight off and recall them,
and this helps us to rewrite

these. Square roots we can write
square root of 400 as the square

root of 4 times by 100.

And the square root of 90 is the
square root of 9 * 10.

Then we expand out.

And we say that this product and
the square root of that product

is the square root of 4 times by
the square root of 100.

Times by the square root of 9
times by the square root of 10.

Then we simplify. We know the
square root of 4 because there's

a square number and it's two. We
know the square root of 100 is

10 square root of 9 is 3.

Thought the square root of 10 is
just the square root of 10.

And we meeting it up by
multiplying these three numbers

together. That's 2 * 10 * 3 is
6060 Route 10 delicious.

What if we had a quotient?

Involving square roots.

What if we had the square root
of 2000 divided by the square

root of 50? Well, we use a
similar rule to the way that we

use the rule for multiplication.
We can say that that is equal to

the square root of 2000 / 50.

We cancelled I'm.

And we get.

That The answer is the square
root of 40, but we can simplify

the square root of 40. Thinking
of square numbers breakdown 40

as 4 * 10 and that equals the
square root of four times the

square root of 10, which is 2
times the square root of 10.

What could be simpler? It
becomes second nature really,

the more practice you do with
the rules, the more you

understand them and.

You don't have to think
about them.

Now, these simplifications of
Surds are quite easy ones.

But what if we had this
expression involving Surds?

It's a little bit different from
the expressions that we've

already used. One plus square
root of 3 times by two minus the

square root of 2. How do we
expand out an expression like

this where we do it just in the
same way as we did normally?

We use the first number in these
brackets, multiply it by these

two numbers, and then we use
this number and multiply it by

these two numbers and combine
them together if possible. So if

we do that, we multiply 1 by
these two numbers and we get 2

minus the square root of 2 and
we multiplied by square root of

3 that is.

Square root of 3 * 2 is 2 square
root of 3 and then the square

root of 3 times negative square
root of 2, which is your

subtract. Route 3 Route 2.

Now when we look at that

expression. And consider it you
can see that we can't really

simplify it any further.

But what if we had this

expression? Similar to the one
above using two brackets, but

look at the numbers and the
3rd's involved. One plus square

root of 3 times by one minus the
square root of 3. The numbers

the same, but the operations
different. See what happens when

we multiply out in the usual
way. 1 * 1 minus Route 3 is 1

minus Route 3, then Route 3
times these two Route 3 * 1.

Is plus 3 route 3 times
by minus 3 three is subtract.

Route 3 route 3.

Now remember. That little fact
that we did previously. This is

Route 3 squared and what's Route
3 squared? It's just three so we

can look at our expression and
we can see that that is one.

And loan behold, look what
happens here in the middle.

Subtract right, three, add Route
3, they cancel.

And so we're left with one

minus. This which is three
1 - 3 is negative 2.

So we started off with a
multiplication of brackets,

which involves surds. But look
what happens with the answer.

We've got a whole number with
no service involved.

Now this is a significant fact,
and in fact this is so well

known it's given a name. This
type of expansion is given a

name. It's called the difference
of two squares.

And the difference of two
squares is written as a squared

minus B squared, and it can be

expanded. By using two brackets.

In two A-B times

by A+B? Now, can
you see that this is similar to

what we had up here, where a is
one and B is Route 3?

So. Our product here. One
plus Route 3 times by one

minus Route 3 is really the
difference of two squares and

the two squares are 1 squared.

Minus Route 3 squared.

Again, notice we're using that
simple fact, which makes things

really easy. That is 1 - 3,
which is negative 2.

So if you remember this

expansion. The difference of
two squares. This is very

helpful sometimes when you
have calculations involving

sired serves.

Night, sometimes you have surged
written like this one over the

square root of 13.

Now really, we don't like
answers being given such that

the denominator is assert.
What we want to do is give an

answer with a whole number as
the denominator.

Now there is a technique that
helps us to change this sired

into whole number and that
technique is called

rationalization. What we have to
do is rationalize the Route 30

we have to make it into a whole
number. Now the easy way to do

that is just use that simple
fact that keeps cropping up time

and time again is that we have
to multiply a thigh itself.

But we don't want to change the
value of this fraction. So what

we have to do is multiply it by.

The fraction square root of 13
over the square root of 13

because that fraction is equal

to 1. Now when we do

that. And evaluated.

The square root of 13 times the
square root of 13 is there Ting

and on top one times the square
root of 13 is the square root of

30. And this is the more
acceptable form of writing one

over the square root of 30.

It's where you have, as I
said before, a whole number

as the denominator.

Now we can get more complicated
fractions, which involves

serves. What if we had this
fraction one over 1 plus the

square root of 2?

Hi can be rationalized that high
can we make this?

Addition into a whole number.

Will remember what we've just
done on the previous examples.

We use the difference of two

squares. And the difference two
squares had an expansion where

you had the two numbers the
same, but the operation was

different. So what if we
multiplied the one plus the

square root of 2 by 1 minus root

2? Now what we've got on the
bottom is the expansion of

the difference of two
squares, and we know that

will give us a whole number.

But we want to leave that this
fraction has the same value, so

we have to multiply it by the
fraction 1 minus root 2.

Over 1 minus root 2 because
that fraction equals 1.

So we evaluate what we've got
now. Remember, this was the

expansion of two squares and one
of the two squares is 1 squared

and the square root of 2.

1 squared minus the
square root of 2.

That's what this
product is when we

evaluate that that
equals 1 - 2.

Which equals negative one.

Now we simplify all of this.

1 * 1 minus square root of 2
is 1 minus the square root of 2

all over negative one.

Now that doesn't look really

neat. What we will do now is
meeting it up. We divide through

by the negative one and we
divide through by the negative

one. We get negative one into
one which is negative one.

Than negative one into the.

Negative Route 2 will give us
Plus Route 2.

And we'll just rewrite it
so that we've got the

positive number. First,
Route 2 subtract 1.

So when we rationalize this
fraction, one over 1 plus the

square root of 2, you get the
square root of 2 - 1 is not

really neat. It looks great.
You've got a fraction here, but

here you've just got a

subtraction. 1

final example. More
difficult than the previous

ones, see how we can cope with
this one over the square root of

5 minus the square root of 3.

We want to rationalize this

fraction. Think of the
difference of two squares.

We've got ripped 5 minus Route

3. If we multiply it by
Route 5 Plus Route 3, we know

that that expansion is the
difference two squares.

We don't want to change the
value of the original fraction,

so we have to multiply it by
another fraction which equals 1

and that will be square root of
5 + 3 over the square root of 5

plus the square root of 3.

We work guys.

This expansion using the
difference of two squares, the

two squares involved are the
square root of 5.

Subtract the square
root of 3.

When we evaluate that that is 5

- 3. Which equals 2.

So when we rewrite all of this
one times, the square root of 5

plus the square root of 3 on
top over 2.

We tally it up a little bit.

By dividing through by the
two and so we get the square

root of 5 over 2 plus the
square root of 3 over 2.

So that's it. That's how you

rationalize. Search You have to.

Make this error denominator into
a whole number.

By multiplying the fraction.

By another fraction,
which is one.

And it's helpful to remember the
difference of two squares.

Now I want to want to do is
finish off by giving you my top

tents on serves. First of all
recap what is assert assert is a

fractional root of a whole

number. Like a square root or
cube root, the gives you an

irrational number and remember
an irrational number is a number

that cannot be written as a
whole number or a fraction.

Remember there are common serves
that we use a lot as route to

Route 3, Route 5 and so on.

And there's certain basic
rules that we know that are

useful when we are doing
calculations involving Surds.

One of them is if you have the
square root of product.

Then that can equal the square

root. Of one number times by the
square root of the other number.

And similarly, the square root
of a quotient.

Is the square root of 1 number
divided by the other square

root? But remember the
square root of addition is

not the sum of the square
roots that a common mistake,

so don't make it.

Another technique that we use

insert calculations. Is

rationalization? We don't like
surge being the denominators of

fractions. So what we have to do
is rationalize the fraction.

We multiply by a fraction that
is equal to 1.

And sometimes when
we have to do that.

The difference of two squares is
useful. The expansion of the

difference of two squares is.

A squared minus B squared is a
minus B Times A+B.

Now, if you know all of those
top tips, I think you'd be

pretty good in doing
calculations involving surds.