Hello everyone, welcome back to a new video In the last
video, as you might have seen, we did indices that is power
related things
If we have any base, if we are multiplying it by n times
then we can write it as a to the power n
In that, n was our indices and what was a? a will be our
base
So now what we are doing is, you can say it is opposite or
related to it
We are doing the next thing that is thirds
Where did the third come from?
In the topic we were doing, we were doing one thing n times
and multiplying the same thing
But what was in our power? a to the power n
But if suppose we get a to the power 1 by n something like
this, means what is power? 1 upon something
So how can we write this thing? We can write this thing
like this
We will put the sign of root like this
You might have read root 2, root 3, irrational numbers
So you will put the sign of root like this
In that you will write a inside, whatever was the number a
and here you will write a small n
In this root, in the leg or whatever hand you can understand
In that you will write n
So what will happen? nth root, what will we call this? nth
root, nth root of what? nth root of a, whatever number is
inside
Like you might have read in your childhood
If we write like this, root 2
What is the meaning of this?
What else we call this?
We call this root, simple
Or we call this square root
Square root
We will call this square root
Because this is power
What else is the meaning of this? 2 to the power 1 by 2 So
square root was like nth root So this is for 2
So square root, what is there in this? 1 by 2
Similarly, if we make the sign of cube root
Sign of cube root, like we have to take the power of 3 in
cube root
So here we write 3
So what is this?
Cube root
Sign of cube root
That is power of 3, 1 by 3
And if I give a simple example If I give a simple example
What will I write? 27 Cube root of 27
So what will that become?
Power of 27, 1 by 3 1 by 3 Now how will we write this?
What will we write the number inside? In this way, we will
write in multiple
Of same number So that our power becomes simplified
Now if we have 1 by 3 in power
So we have to subtract 3 from below
We cannot solve that
So to subtract that, we need a number inside
Whose power is 3
So what can we write 27? 3 to the power 3
We can write this as 3 into 3 into 3
And this whole power 1 by 3 So what will we write 3 into 3
into 3?
We have seen in the last video 3 to the power 3
Now you will apply the same property
That if a to the power m Whole power is n
Means there are two powers in whole So what will happen?
Our power will multiply
So if we multiply this also
What will 3 into 1 by 3 become? 3 by 3 will cancel out
So what will be the answer?
Simple 3 to the power 1
That is 3 As we have studied anything raised to the power 0
Anything raised to the power 0 is 1
Similarly, anything raised to the power 1
Is that number only
Any number raised to the power 1
Or that number as it is written
They both are same
Will be same
Now see if you understand this much
So on this root
We will write this small number
Like this is cube root
This is square root
Similarly, if this is any other number
Suppose it will be 9
Then 1 by 9 will be the power
And we will write 9 on its leg
In the leg of the root
Then it will be 9th root
Cube root
Then it will be 9th root
Like cube, square root
Now there are some similar properties
Like we have done in indices
Similar properties
Similarly analogous
We will get in this also
In the surds
Like The main thing you remember
That if this is under root
N In power bar
And a is there
So how can we simplify it a raised to the power 1 by n
Write the properties
First This is the main point
First Next Suppose This is under root
N
How can we write this
Same a to the power 1 by n
If we have a to the power n
If we don't have a a to the power n
How can we write this
Like I have shown you last time
On the same note
We will write
Like multiply the powers
First simplify it
Under this nth power
How can we write a to the power n
Whole power 1 by n
Like this
Now multiply the powers
Rule of indices a raised to the power n by n
Cancel out
That is the answer
We have a
Next property
Property number 3
Suppose Under root
Whole a Into another number b This is in the same root
In the nth root
How can we simplify this
We can write this in another form
Like a into b to the power
Whole power m
I have shown you this property
In another way
If a to the power m
Into b to the power m
We can write a b whole power m
You can read this
In reverse
The problem is
If two numbers are multiplying
And their powers are same
What can we do
Multiply them
And distribute the powers
On the same page
We will do this
Like these two numbers are multiplying
In the nth root
Separate them
Distribute them
This will be a to the power 1 by n
If both are multiplying b to the power 1 by n
What happens in both
In the nth root
Distribute If we write a to the power n
Into b to the power n
This is same
If we solve this a to the power 1 by n
If we solve b b to the power 1 by n
Both are multiplying
Multiply This is one property
Next property
If you see here
This property
Was same
What was the nth root
It was same
Suppose we have a to the power
Root m
Inside is root m
Outside is root n n This is the question
How to solve this
This property will be simple
Don't do anything
Step by step
Simplify First of all
We simplify
This thing inside
What can we write
Simplifying this a to the power 1 by m
Can we write
We replace this
Inside thing
Then write outside
Root n
Then again
Consider this as one number
Inside and simplify
What will be the power 1 by n a to the power 1 by m
What will be the power 1 by n
Under root n
Now see
Apply the property a to the power m
Whole power n
Then what will happen
Powers will multiply
Multiplying these two
What will be the result a to the power 1 by m
Into n
Now 1 by anything
If the power comes
Then what will be the result the nth root is m n n square
is the nth root is the nth root is the nth root is