Hello everyone, welcome back to a new video.
In today's video, we are going to do indices and surds.
What are indices and surds?
First of all, if we have a number 2 plus 2 plus 2 plus 2.
2 plus 2 plus 2 plus 2.
I have added 2 four times.
So, what else can we write?
If we write 4 into 2, we will get the same answer.
If we multiply something by its number of times, we will
get the same answer.
Similarly, if we are multiplying a particular number by its
number of times, we will get 2 into 2 into 2 into 2.
In this case, we have added 2 four times, so we get 4 into
2.
But, we are multiplying by its number of times.
So, we will write 4 into 2.
What does it mean?
It means that we are multiplying 2 by its number of times.
Similarly, if we have a particular number,
2 is the base of this number.
4 is the index of this number.
The power of this number will be the index of this number.
The power of this number will be the base of this number.
Did you understand this?
Now, if we generalize this, that is, if we have a
particular number, it could be any number.
It could be a natural number, a fractional number or any
other number.
So, we multiply a into a into a.
We multiply a into a by its number of times.
If we multiply a into a by its number of times, then we can
write a to the power n.
Here, what is the base?
The base is a and the index is n.
n is the power.
So, we will call it a.
So, in the topic of search and indices, the first index is
a.
Now, there are some properties of this.
If we are multiplying numbers in the form of power, then
what are their properties?
So, you write that.
Suppose we have a to the power m into a to the power n.
This m and n can be same or different.
But what should be the base a?
It should be same.
So, what will happen?
If we are multiplying numbers, then what will be the powers?
We will get powers.
We will get add.
If we are multiplying numbers with their powers, then what
will happen?
If we are multiplying numbers with their powers, then what
will be the powers?
We will get powers.
We will get add.
Suppose, we have a to the power 2 into a to the power 3.
If I expand it, then we can write it as 2 into 2 2 to the
power 2 and 2 to the power 3.
2 into 2 into 2
Now, what are the 5 2's?
They are multiplying each other.
So, we can write them as 2 to the power 5.
What we are doing is that we are multiplying the same
number with the same power.
So, we can write it as 2 to the power 5.
If we apply the Givali property and rule, then what will
happen?
It is the same base.
The base is the same.
So, both the bases are the same.
So, what should we do?
We should add the powers.
So, 2 to the power 2 plus 3 will be added.
So, what is the answer? 2 plus 3 is 5.
So, 2 to the power 5.
So, we got the same answer from this property.
And the answer we wrote in the form of expanding is also 2
to the power 5.
So, this is the property number 1.
When the base is the same but the power can be equal.
If it is equal, then what will happen?
It will be n plus n.
And it can be different also.
That is up to the question.
But if it is in multiplication, then what will be the power?
The power will be added.
So, let's see what is the next property?
Suppose, we saw that the base was equal and the base was
same.
Suppose, this time the base is not the same but the indices
are the same.
That means the powers are the same.
Suppose, we have one number a whose power is m and the
other number is b raised to the power n.
b raised to the power
Sorry, the powers should be the same.
So, we will also write m.
What is the power of both?
It is the same number.
It is in the power of a and in the power of b.
So, how can we do this?
We can write this.
What can we do for both of them?
We can take the power common.
How can we take the common?
In common, first we can write a into b and what can we do
for both of them?
We can write the whole power of both.
If the powers of both are same, if powers are same,
then we can do this.
But we cannot do this when the powers of both are same.
What was the last rule?
When the bases of both are same.
This is the power of same.
Let us see an example of this.
Suppose, this is the power of 4 which is 3 into the power
of 7 which is 3. Now, we can take this out as 4 cube into 7
cube.
We can take this out separately
or we can write it directly as 4 into 7 and what can we do
for the whole power of it?
We can write it as whole cube.
So, what will we get?
We will get the same answer.
The answer of both of them will be the same.
Now, this is the same power.
Next property that you are getting is that if we have the
power in division,
that is, in denominator, then we have 1 upon a power m.
1 upon a power m.
So, in what form can we write it?
We can write it in this way also.
If the power is in denominator then we have to write the
power in numerator instead of denominator.
So, what happens?
If the power is in denominator, then we can write it as
negative in numerator.
Suppose, 1 upon a power m is there, then we can write it in
another form that is, a power minus m.
This is a to the power minus m.
We can write it in this way.
If we can write it in this way, then you guys tell me how
to write it.
a to the power p upon a to the power q.
Solve it and tell me how to write it with the help of this
result and the help of the first result.
Do it.
First of all, see this.
How can we change it?
The power in denominator will go up and become negative.
So, we can write it as a to the power p into a to the power
minus q.
Why? Because this power has gone up.
If we write it in numerator, then it will become negative.
Now, see the first property will be applied.
Why? Because we have the same base.
The base of both is a.
That is the same base.
When the base is same and both are in multiplication, then
the power will add up.
result. Fourth result, you write it as a to the power p
divided by a to the power q would be equal to a to the
power p minus q.
Whose help did this come from?
It came from the help of rule number 3 and rule number 1.
So, similarly if I tell you another case of this,
If we have this, a to the power m and b to the power m.
How will we write this?
Now, when I made you do the same thing in multiplication,
So, multiply both of them together and then do the whole
power.
So, it will give the same result that you get different
powers.
So, similarly what will happen in this case also?
If these two are in divide and we have the same power.
So, we don't have to do anything.
We take them as common first.
Power a by b and their whole power m.
Whole power m.
So, this is your rule number, that was 4.
So, this is your rule number 5.
If we have this given, a b to the whole power m.
So, we can write this in another form.
That is, we can write this in another form.
We can write this as b by a to the power minus m.
b by a to the power minus m.
Why minus m? Because we have done reciprocal inside.
And when we do reciprocal inside,
Then our power will become negative.
Because we are doing the opposite case.
When we did reciprocal inside,
For example, we did reciprocal inside 1 by a to the power m.
What did we do?
We can write this as 1 by a to the whole power m.
Because any power of 1 will be 1.
So, we can write this as 1 by a to the power m.
Now, if we do reciprocal inside.
Apply this rule.
If we do reciprocal inside,
Then our power will become negative.
So, if we do reciprocal inside.
Reciprocal means,
Write denominator in numerator and numerator in denominator.
That will be called reciprocal.
So, this is a by 1 to the whole power.
What will be the negative?
So, minus m.
This is visible, right?
This will become minus m.
So, anything divided by 1,
We can write it as 1 by a to the power minus m.
This is our property.
We can write 1 by a to the power m.
Similarly, this property is based on it.
That is 6.
What is this? If we have a by b to the power m.
We can write it as reciprocal in negative power.
That is b by a to the power minus m.
Next, write 7.
7 is a very cute property.
All the kids must know this.
If we have any number.
x, y, z, any number.
We don't care about it.
But, suppose that number is a.
If we have any number.
What will be the power in index?
It will become zero.
That means, power will become zero.
Then, we will get only one answer.
If we make any number raise to the power zero.
Then, we will get only one answer.
Similarly, we have done the multiplication result.
If we do something else.
Like, we were multiplying only two numbers.
Similarly, we can multiply three numbers.
a to the power m into b to the power m into c to the power
m.
How will we do that?
Suppose, look at the corollary.
If it is not necessary.
If it is two only.
It can be more than two.
We don't have to do anything.
We have to apply the same rule.
It was a simple rule.
If it is the same power.
Then, multiply it.
And, take the common power.
Similarly, if it is in three.
It will be in three.
But, what should happen?
Power should be same.
Then, multiply three inside.
And, take the common power m.
So, the rule is applied in three also.
3, 4, 5, 10, 15.
In any number.
In finite number of times.
This rule will be done.
It will be applicable.
Note down this much.
So, now moving on to next property.
How far we have done?
We have just noted this.
This is the sub-case.
Next is done.
It is done till 7.
Now, you write 8th property.
What is 8th property?
Suppose, we have a to the power m.
And, in this whole power is n.
Means, first we are writing 2 to the power 3.
Then, after solving this.
We are taking out the whole power 5.
This.
So, this will be equal to.
This will be equal to a to the power m into n.
We can do this.
The powers are multiplied.
So, if this is 2 to the power 3 into whole power 5.
We can write this as 2 to the power 3 into 5.
That is 2 to the power 15.
We can simply write this and solve.
If I say one more case of this.
Its sub-case.
If we have this power instead of n.
It becomes 1 upon n.
This power becomes 1 upon n instead of n.
So, nothing will happen.
The rule will be same.
That whole power is in this.
So, powers will be multiplied.
So, multiply these two powers.
So, this will be a to the power.
Multiply this.
So, this will be m upon n.
Because, this is in denominator.
So, n in denominator.
And, m into 1 in numerator.
So, this will be m.
So, this will be a to the power m into n.
This is its sub-part.
This is its corollary.
Next point.
Which number was this?
Point number 8.
Point number 9.
You must have read this.
Let me tell you one more case of this.
You have this.
a to the power m whole power n.
Suppose, you have this.
If I tell you another case of this.
If you have this.
a to the power m whole power 1 by m.
Note down this difference.
Because, this is very important.
If you can find the difference between these two things.
Then, it will be easy for you.
How will you solve this?
Will you write this as a to the power.
Ma'am said, if it is like this, then multiply the power.
Will you write this as a to the power?
You cannot write this as a to the power.
Why can't you write this?
What is the difference between this and the last one?
In the last one, I have put a bracket.
After putting a bracket,
This whole power is 1 by n.
Suppose, if we had this.
Suppose, we had this.
2 to the power 3 whole power 1 by 6.
Suppose, we had this.
We multiply 3 into 1 by 6.
We get 2 to the power 3 by 6.
That is 2 to the power 1 by 2.
This is your answer.
But, if this is the same thing in another way.
If I write this as 2 to the power 3.
And, 3 to the power 1 by 6.
In this one, 2 to the power 3.
This whole power is 1 by 6.
But, in this case, only the first power is 3.
Its power is 1 by 6.
So, these two things will not be equal.
Because, after solving this, what is coming? 2 to the power.
First, we will solve 3 to the power 1 by 6.
Then, we will solve 2 to the power 1 by 6.
So, these two cases are completely different.
equal to a to the power of x, b to the power of x.
If we have given equal to both the quantities, then the
base will also be equal.
If we have given equal to both the quantities, then the
base will also be equal.
Similarly, if we have given equal to both the quantities,
then the base will also be equal.
Now, suppose we have given a to the power of p is equal to
a to the power of q.
This is given to us.
So, this implies that the bases are same, but we do not
know the power.
And both the quantities are equal.
So, this implies that the power will also be equal.
This is called equating the powers.
In this, the powers will be same.
Where do these things get used?
Suppose these things get used.
You are given that 2 to the power of x is equal to 2 to the
power something 16.
Find the value of x.
So, what happens in this?
It becomes same.
If the bases are same, then the power should also be same.
Because both the quantities are equal.
So, this implies that x is equal to 16.
This was simple.
Suppose the power is x minus y.
On the left hand side, the power is x minus y.
Then equate the powers.
Now, x minus y is equal to 16.
Such equations are formed.
When we have to solve the questions, then such equations
are formed.
By equating the bases and powers, we further solve the
questions.
And find out the unknown quantities.
These were the 9 properties and their subparts.
All these are the properties and rules of indices.
In the next video, we will do a different case related to
the third.
Thank you for watching the video.
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