Subtracting and Simplifying Radicals

u16_l2_t2_we2 Subtracting and Simplifying Radicals

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.