More rational exponents and exponent laws | Algebra I | Khan Academy

More Rational Exponents and Exponent Laws

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Simplify the expression
using rational exponents.

So we have the cubed root of
all of this stuff over here,

which would be the
same thing, or which

is the same thing by definition,
as all of this stuff--

64a to the sixth, b to
the third, c to the ninth,

raised to the 1/3 power.

These are equivalent
by definition.

And if we take the product
of a bunch of things

and raise them to
an exponent, that's

the same thing as raising
each of these things

that we're taking the product
of to the exponent first

and then multiplying.

So this is going to be equal
to 64 to the 1/3 power times

a to the sixth to
the 1/3 power times

b to the third to
the 1/3 power times

c to the ninth to the 1/3 power.

And we've seen this in
several examples already.

Well first let's try
to simplify this.

64 to the 1/3.

You might recognize
that already,

but we can factor 64
as-- well, I wouldn't

do a pure prime
factorization, because you'll

see it's 4 to the third power.

Actually, you could
verify that for yourself.

Or I encourage
you, if that's not

obvious to see that this is
4 to the third power, 64,

multiply it out for yourself.

Or if you're like,
how did you know that?

Do the prime factorization.

And what you're going
to do is if you--

well, let me just show you.

If you do the prime
factorization,

you're going to have 2 times 32.

32 is 2 times 16.

16 is 2 times 8.

8 is 2 times 4.

And then 4 is 2 times 2.

So you can have 2 multiplied
by itself three times,

or you could have it-- a 2
multiplied by itself three

times and that happens twice.

Or you could have
4 times 4 times 4.

So if it immediately
didn't jump into your head

that 64 is 4 to the
third power, you

can literally do a brute
force prime factorization here

and see that you have
6-- you could factor this

into 2 to the sixth power,
or the same thing as 4

to the third power.

So I'll let you think
about that a little bit.

But this right here simplifies
that 64 is 4 to the third power

and you raise that
to the 1/3 power.

That is the same thing.

Actually, let me
write it that way.

That could make it
more interesting.

So this right here, this
is 4 the third power.

So I have 4 to the
third power and then

that raised to the 1/3 power.

So if I want to rewrite
that term right over here,

that's going to be
four to the 3/3 power.

I'm literally
multiplying 3 times 1/3.

4 to the third and
then that to the 1/3.

Then over here, I have a to
the sixth and then that the 1/3

power.

That is going to be a to
the 6 times 1/3 power,

or a to the 6/3 power.

Then I have b to the third and
then that raised to the 1/3,

so that's going to
be-- we literally

can multiply exponents here.

So we have b to
the 3 over 3 power.

And then finally, we
have c to the ninth

and then that raised
to the 1/3 power.

So that is c to the 9/3 power.

And if you simplify it, this
first term here, 4 to the 3/3,

that's just 4 to
the first power.

So that's just 4.

We have a to the 6/3.

6/3 is just 2, so that
just becomes a squared.

B to the 3/3 power, that's
just b to the first or b.

And then finally, c
to the 9/3, that's

just c to the third power.

So we write c to the third here.

And we are done.