Simplify 2a squared over 3b to the fifth,
all of that to the fourth power.
So here we can use the property that if I have x over y,
and I'm raising that to the nth power,
this is the same thing as x to the nth power over y
to the nth power.
And we could see that with an example real fast.
If I had, let's say I had x over y to the third power,
what would this be equal to?
This would be x over y times x over y times x
over y, which is going to be equal to x times x
times x over y times y times y.
And this is clearly x to the third over y to the third.
So I just gave you a particular example of why this works.
Hopefully, it gives you enough confidence
or makes you feel good about using this property.
So let's use it right here in our actual problem.
So we have 2a squared over 3b to the fifth,
all of that to the fourth power.
So this is going to be the same thing as 2a
squared to the fourth power.
So let me write it this way.
It's going to be 2a squared to the fourth power.
Let me do the fourth power in blue.
All of that over 3b to the fifth to the fourth power.
And now we can use the property that if I
take the product of things and raise that
to a power, that's the same thing as raising each
of the things in the product to the power
and then taking their product.
So this is going to be the same thing.
Let me just write the property here.
So if I have x times y to the nth power,
this is the same thing as x to the nth times
y to the nth power.
And you could do a similar argument
that we did right here to see why this works out.
So let's do that over here.
2a squared to the fourth power is going to be equal to 2
to the fourth power-- I'll color code
it-- 2 to the fourth power times a squared to the fourth power.
And then all of that over 3b to the fifth to the fourth power.
That's going to be 3 to the fourth power times b
to the fifth to the fourth power.
Now, the last interesting thing here-- and we
can evaluate what these numbers come out to be.
2 to the fourth power-- let me just do it right now.
2 to the fourth-- 2 squared is 4.
2 to the third power is 8.
2 to the fourth power is 16.
So this right here is 16.
3 to the first power is 3.
3 squared is 9.
3 to the third power is 27.
3 to the fourth power is 81.
So this over here is-- let me do it
in that green color-- this is 81.
And to simplify these terms here,
we just have to remember the--I guess we call it the power
property or the power rule of exponents.
And that's just the notion that if I have x to the n
and then I raise that to the mth power,
this is the exact same thing as x to the n times m power.
So if we wanted to simplify this,
our numerator is now 16-- that's the 2 to the fourth--
and then a squared and then that the fourth power.
That's going to be a to the 2 times 4 power,
or a to the eighth power.
This is literally 2 times 4, so we're just
multiplying 2 times 4.
This 2 times this 4 to get that 8.
And then in the denominator, we have our 81 and then
b to the fifth and then that the fourth power.
That's going to be b to the 5 times 4 power or b
to the 20th power.
So I got 20 by multiplying 5 times 4.
And we're done.
This is about as simplified as we can get.