Assalamualaikum warahmatullahi wabarakatuh. In this video we will be discussing on the rules of indices.
The first rule that i'm going to
introduce
is called the zero index. Zero index
means
that any base with power zero
is always equal to 1. For example, here
I have
m^0. This is equal to 1.
The same goes with 4^0. 4
is the base and 0 is the power. This is
equal to 1.
The next one is negative index. Now
negative index
means reciprocal. Yes, that is a very big
word.
I'm going to write it down and explain
that word first for you. The word is
reciprocal. Reciprocal means 1
over something, so when we have a^(-m)
We can write this down as its
reciprocal
that is 1 over a^m.
So, let's see the example here. In this
example we have
a^(-3), so the
reciprocal of that
is 1/(a^3).
On the next one we have 3^(-2). Since the base 3
has negative index -2. This
is equal to its reciprocal which is (1/3)^2
and if we
calculate this
this is equal to 1/9. The third one
is fractional index when the power is in
fraction. For example here 1/m. This will
become its root. So here we have
a^(1/n). This
is equal to the nth root of ‘a’. Let's look
at the example here.
We have the square root of ‘y’. So to write
this in
an index form, ‘y’ as the base and we see
2 as its root. So we can write the power
as y^(1/2).
Let's see if you can get the next one
correct. You can pause here to figure it
out first. I'm going to proceed with the
answer. So it's 7^(1/3). 7 is the base with fractional
power 1/3. Therefore, we can write
this
as the cube root of 7. Let's proceed to
the next slide
before I proceed with the next rule you
need to remember
that the rules of indices can only be
applied on the same base. So
the fourth row, here I have a^m
multiplied by a^n. This
is equal to a^(m+n).
You can ask your lecturers in
tutorial classes
to prove this rule for you. Look at the
example here
I have x^3 multiplied by
x^4.
Since the base are equal so this will
become x^(3+4).
and this is equal to x^7.
I'm going to leave the second example
here for you to do.
The next rule is when we have a^m
divided by a^‘n
and this is equal to a^(m-n).
Again you can ask your tutorial
lecturer to prove this one for you.
So in the example here I have
x^8 divided by x^4
So according to this rule I can write
this as
x^(8-4) and this
is equals to x^4.
Number 6
when I have a^m and the
whole thing
to the power of ‘n’. This will become a^(mxn).
Therefore for this example here
I have x^3 and the whole thing
to the power of 5. This will equal to
x^(3x5).
So this is x^15.
You can pause the video here if you need
more time to find the answers.
For the second example here, let's check
the answers.
The first one is 2^10
the second one is 5^7 and the last one is 4^(3/2).
I hope you got it all right you can high
five your friend next to you.
Don't forget your hand sanitizer though.
Okay so the last two rules
is for us to know when to distribute the
power since the definition of an index
is from multiplication
that means only multiplication and
division will be affected.
This is also means you cannot distribute
power if the operation
is + or - . So if you have
a multiplied by b and all this to the
power of m.
You can distribute its power so it'll be
a^m ;  b^m.
Look at the example here you have
(xy)^5 because the operation
for x and y
is multiplication we can distribute its
power to become x^5
and y^5. The same goes
for the next one
because its operation is division. Here two
we can distribute its power.
So in the example here we have
w/z so you can distribute its
power
to become (w^4) / (z^4). You can pause here to
take some times to answer
these two examples. Let's check the first
one is
2^4 ; w^4 ; z^4
If you
calculate this
you should get 16(w^4)(z^4). The next one would
be 4^3 divided by 7^3.
If you calculate this
you should get 64/343.
So that's it for the definition and the
rules of indices.
If you are still confused or has any questions,
you can watch the video again or you can
ask your lecturers
during your tutorial sessions you can
also ask your question
on the comment section below we will try
to clear your confusion
as soon as we can. The next video will
explain
these two questions make sure to answer
them first before you check your answers
and understanding on this subtopic.
Thank you for your attention
see you on the next video. Take care and goodbye.
