Exponent Rules (Part - 2) - Math, Class 10 Video Lecture

Welcome to Part 2 on the
presentation on Level
1 exponent rules.
So let's start off by
reviewing the rules
we've learned already.
If I had 2 to the tenth times 2
to the fifth, we learned that
since we're multiplying
exponents with the same base,
we can add the exponent, so
this equals 2 to the fifteenth.
We also learned that if it was
2 to the tenth over 2 to the
fifth, we would actually
subtract the exponents.
So this would be 2 to
the 10 minus 5, which
equals 2 to the fifth.
At the end of the last
presentation, and I probably
shouldn't have introduced it
so fast, I introduced
a new concept.
What happens if I have 2 to
the tenth to the fifth power?
Well, let's think about
what that means.
When I raise something to the
fifth power, that's just like
saying 2 to the tenth times 2
to the tenth times 2 to the
tenth times 2 to the tenth
times 2 to the tenth, right?
All I did is I took 2 to
the tenth and I multiplied
it by itself five times.
That's the fifth power.
Well, we know from this rule up
here that we can add these
exponents because they're
all the same base.
So if we add 10 plus 10
plus 10 plus 10 plus
10, what do we get?
Right, we get 2 to
the fiftieth power.
So essentially, what
did we do here?
All we did is we multiplied
10 times 5 to get 50.
So that's our third exponent
rule, that when I raise an
exponent to a power and then I
raise that whole expression to
another power, I can multiply
those two exponents.
So let me give you
another example.
If I said 3 to the 7, and all
of that to the negative 9, once
again, all I do is I multiply
the 7 and the negative 9, and
I get 3 to the minus 63.
So, you see, it works just as
easily with negative numbers.
So now, I'm going to teach you
one final exponent property.
Let's say I have 2 times 9,
and I raise that whole thing
to the hundredth power.
It turns out of this is equal
to 2 to the hundredth power
times 9 to the hundredth power.
Now let's make sure
that that makes sense.
Let's do it with a
smaller example.
What if it was 4 times
5 to the third power?
Well, that would just be equal
to 4 times 5 times 4 times 5
times 4 times 5, right, which
is the same thing as 4 times 4
times 4 times 5 times
5 times 5, right?
I just switched the order in
which I'm multiplying, which
you can do with multiplication.
Well, 4 times 4 times 4,
well, that's just equal
to 4 to the third.
And 5 times 5 times 5 is
equal to 5 to the third.
Hope that gives you a good
intuition of why this
property here is true.
And actually, when I had first
learned exponent rules, I would
always forget the rules, and I
would always do this proof
myself, or the other proofs.
And a proof is just an
explanation of why the rule
works, just to make sure
that I was doing it right.
So given everything that we've
learned to now-- actually, let
me review all of
the rules again.
If I have 2 to the seventh
times 2 to the third,
well, then I can add the
exponents, 2 to the tenth.
If I have 2 the seventh over
2 the third, well, here I
subtract the exponents, and
I get 2 to the fourth.
If I have 2 to the seventh to
the third power, well, here
I multiplied the exponents.
That gives you 2 to the 21.
And if I had 2 times 7 to the
third power, well, that equals
2 to the third times
7 to the third.
Now, let's use all of these
rules we've learned to actually
try to do some, what I would
call, composite problems that
involve you using multiple
rules at the same time.
And a good composite problem
was that problem that I had
introduced you to at the
end of that last seminar.
Let's say I have 3 squared
times 9 to the eighth power,
and all of that I'm going to
raise to the negative 2 power.
So what can I do here?
Well, 3 and 9 are two separate
bases, but 9 can actually
be expressed as an
exponent of 3, right?
9 is the same thing as
3 squared, so let's
rewrite 9 like that.
That's equivalent to 3 squared
times-- 9 is the same thing as
3 squared to the eighth power,
and then all of that to the
negative 2 power, right?
All I did is I replaced 9
with 3 squared because
we know 3 times 3 is 9.
Well, now we can use the
multiplication rule on
this to simplify it.
So this is equal to 3 squared
times 3 to the 2 times 8,
which is 16, and all of
that to the negative 2.
Now, we can use the first rule.
We have the same base, so we
can add the exponents, and
we're multiplying them, so that
equals 3 to the eighteen power,
right, 2 plus 16, and all
that to the negative 2.
And now we're almost done.
We can once again use this
multiplication rule, and we
could say 3-- this is equal to
3 to the eighteenth times
negative 2, so that's
3 to the minus 36.
So this problem might have
seemed pretty daunting at
first, but there aren't that
many rules, and all you have to
do is keep seeing, oh, wow,
that little part of the
problem, I can simplify it.
Then you simplify it, and
you'll see that you can keep
using rules until you get
to a much simpler answer.
And actually the Level 1
problems don't even involve
problems this difficult.
This'll be more on the
exponent rules, Level 2.
But I think at this
point you're ready
to try the problems.
I'm kind of divided whether I
want you to memorize the rules
because I think it's better to
almost forget the rules and
have to prove it to yourself
over and over again to
the point that you
remember the rules.
Because if you just memorize
the rules, later on in life
when you haven't done it for a
couple of years, you might kind
of forget the rules, and
then you won't know how
to get back to the rules.
But it's up to you.
I just hope you do understand
why these rules work, and as
long as you practice and you
pay attention to the signs,
you should have no problems
with the Level 1 exercises.
Have fun!