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.