We're asked to subtract all of this craziness
over here, and it looks daunting, but if
we really just focus, it actually should be
pretty straightforward to subtract and simplify this thing.
Cuz right from the get go, I have 4 times the fourth root of 81x to the fifth,
and from that I wanna subtract 2 times the fourth root of 81x to the fifth.
And so, you really can just say, lookI have four of something and this
something, I'll just circle in yellow I have four of this, it could be lemons.
I have four of these things and I wanna subtract two of these things.
These are the exact same things.
They're the 4th root of 81x to the 5th, 4th root of 81x to the 5th.
So if I have four of, if I have four lemons and
I wanna subtract two lemons, I'm gonna have two lemons left over.
Or if I have four of this thing and I take away
two of this thing, I'm gonna have two of these things left over.
So these terms right over here simplify to 2
times the fouthth root of 81x to the fifth.
And I got this 2 just by subtracting the coefficients 4
something minus 2 something is equal to 2 of that something.
And then that, of course, we still have this minus the regular
principal square root of x to the third, of x to the third.
Now I wanted to try to simplify, I wanna try to
simplify what's inside of these under the radical signs, so that
we can, on this, in this example actually take the fourth
root and over here actually take, maybe, a principal square root.
So first of all, let's see if 81 either is a, is something to the fourth power,
or at least can be factored into something
that is a, a something to the fourth power.
So 81, if we do prime factorization, is 3 times 27,
27 is 3 times 9, and 9 is 3 times 3.
So 81 is exactly 3 times 3 times 3 times 3.
So 81 actually is 3 to the fourth power which is
convenient, cuz we're gonna be taking the fourth root of that.
And then x to the fifth we can write as a product.
We can, let me write it over here so it doesn't get messy.
So I'm gonna write what's under the radical as 3 to
the fourth power, times, times x to the fourth power times x.
x to the fourth times x is x to the fifth power.
And, I'm taking the fourth root of all of this.
And, taking the fourth root of all of this,
that's the same thing as taking the fourth root
of this and, taking the fourth root of this,
and let me just, I'm gonna just skip step.
So, I'm taking the fourth root.
I'm taking the fourth root of all of it, right over there.
And of course I have a 2 out front.
And then x to the third can be written as x squared times x.
It's minus the principle square root of x squared times x.
And I broke it up like this cuz this, right over here, is a perfect square.
Now, how can we simplify this a little bit?
And you're probably getting used to the pattern.
This is the same thing as the fourth root after you get the fourth,
times the fourth root of x to the fourth, times the fourth root of x.
So let's just skip straight to that.
So what is, what is the fourth root?
Well I ca, I can write it, let me write
it explicitly, although you wouldn't have to necessarily do this.
This is the same thing as the fourth, as the fourth root of 3 to the fourth,
times the fourth root of x to the fourth, times the fourth root
of x, times the fourth root of x, and 2 as being multiplied times all of that.
And then this over here is minus the principle square
root of x squared, times the principle square root of x.
And so, if we try to simplify it, the fourth
root of 3 to the fourth power is just 3.
So, we get a 3 there, the fourth root of x to the fourth power is just going to be
x, is just going to be is just, actually, look,
I just reminded myself, you have to be careful there.
It is not just x, because what if x is negative?
If x is negative, then x to the fourth power is going
to be a positive value, and when you take the fourth, remember, this
is the fourth principle root, you're going to get the positive version
of x, or really, you're going to get the absolute value of x.
So here you're going to be getting, you're
going to be getting the absolute value of x.
And then, although, well you could make an argument
that x needs to be positive if this thing
is going to be well-defined in the real numbers,
cuz then what's under the radical has to be positive.
But let's just go with this for right now.
And then we have the fourth root of x.
And then over here the, the principal square of
x squared by the same logic, by the same logic
is going to be the absolute value of x and
then this is just the principal square root of x.
So, let's multiply everything out.
We have 2 times 3 times the absolute value of x.
So, 2 times 3 is 6 times the absolute value of x, times the principal
or the, the principal fourth root of x, I should say, minus,
minus, we've took out the absolute value of x, times the principle root of x.
And we can't do anymore subtracting.
Just because you have to realize, this is a fourth
root, this is a regular square root, principle square root.
If these were the same root, then maybe we could simplify this a little bit more.
And so then we are all done, and we have fully simplified it.
And if you make the assumption that this is defined
for real numbers so that the domain over here this would
has to be under these radicals has to be positive
actually everyone of these cases and if there need to be
positive not gonna be dealing with imaginary numbers all of
these need to be positive, their domains are x has to
be greater than or equal to 0 then you could assume
that the absolute value of x is the same as x.
But I will just take it right here, if you restrict
the domain you could get rid of the absolute value sign.