The radical expression
is not in simplified form because it has a
radical in the denominator.
To write the radical expression in simplified form, we must rationalize its
denominator. That is, we rewrite the expression to make the denominator a
rational number and eliminate the radical.
To do this, we multiply
by 1 written in the
form .
Multiply the numerators. Multiply the denominators.
Simplify the denominator.
The denominator has been rationalized since it no longer contains a
radical. The radical expression is now in simplified form.
Example
Simplify:
Solution
In the numerator and denominator,
cancel the common factor, 2.
Simplify the radical by removing the
perfect square factor, 9.
Simplify
To eliminate
in the denominator,
multiply the expression by 1 written in
the form
.
Multiply the numerators. Multiply the
denominators.
Simplify
and multiply the result by
3.
So,
Note:
is in simplified form.
We cannot cancel the x’s since the x in the
numerator is under the radical symbol,
while the x in the denominator is not. The
same is true for a factor of 2.
is not in simplified form because it has a
radical in the denominator.
.
in the denominator,
multiply the expression by 1 written in
the form
.
and multiply the result by
3. 
is in simplified form.
We cannot cancel the x’s since the x in the
numerator is under the radical symbol,
while the x in the denominator is not. The
same is true for a factor of 2.