When simplifying rational expressions,
it's important to remember
we cannot simplify
across addition or subtraction.
For example here, we cannot
simplify the three and the nine,
because we cannot simplify
across the subtraction.
When simplifying a rational expression,
we simplify the numerator
and denominator completely
then perform the division
indicated by the fraction bar.
If it's helpful, we can think
about having parenthesis
around the numerator and denominator.
To begin simplifying the numerator,
we need to simplify
the square root of 243.
To help us simplify
the square root of 243,
let's look at the prime
factorization of 243.
If we sum the digits,
notice we have two plus four plus three
which is nine, because the similar digits
is divisible by three, so was 243.
Which means 243 is equal
to three times some number.
If we cannot determine this number,
we can always divide 243 by three.
And let's go ahead and do that.
If we 243 divided by three,
we first determine how many
threes in 24, which is eight.
Eight times three is 24,
subtract the difference is zero.
Bring down the next digit,
determine how many threes
in three which is one.
One times three is three,
subtract the difference is zero.
So, because the quotient is 81,
we know 243 is equal to three times 81.
And because 81 is equal
to nine times nine,
and therefore it's a perfect square,
we can actually stop
here and we don't have
to determine the entire
prime factorization.
We now know the square
root of 243 is equal
to the square root of nine
times nine times three.
If we want the square
root of nine squared times
the square root of three which is equal
to nine square root of three.
So, the numerator simplifies
to three minus nine square root three,
this is all divided by nine.
Now, from here there's
two ways to simplify this.
Because we have simplified
the numerator completely,
and we're dividing by
one term or a monomial,
one method is to divide each term
in the numerator by the denominator.
Which means this expression is equivalent
to 3/9 minus nine square
root three divided by nine.
And now simplifying 3/9 simplifies to 1/3,
minus nine divided by
nine simplifies to one.
Therefore, we just have
minus square root three.
So, this would be one way to
simplify the given expression.
Let's take a look at a second method.
Going back up to this expression here.
We got the quantity three
minus nine square root three,
divided by nine.
We look at the numerator
and notice how the two terms
do have a common factor of three.
Let's factor a three from the numerator.
We factor three from the numerator,
we have three times the
quantity one minus three
square root three and all
this is divided by nine.
Of course, we can distribute
here to see we still have
three minus nine square root three.
And now because we have our product here,
we can simplify the
three and the nine here.
There's one, three and three,
and three threes and nine.
Which gives us the quantity
one minus three square
root three divided by three
which is also considered a simplified form
of the given expression.
Notice how the expressions
do look different,
but they are equivalent.
To show they are equivalent,
let's take this expression
and divide each term and the numerator
by the denominator of three.
If we do this we'd have
one divided by three,
minus three square root
three divided by three.
And notice in this form, we
can simplify three divided
by three simplifies to one here.
Giving us 1/3 minus square root three.
So again, these two
expressions are equivalent
and both are considered simplified.
It's important to be aware of this
because when using the quadratic formula
we'll be able to see
expressions in this form.
It's important that we know
how to simplify them correctly.
Either of these methods are valid.
