Section 7.4 Radical Expressions in Simplest Form and Division
A radical expression is in simplest form if:
1. The radicand contains no factor greater than 1 that is a perfect square
(or perfect cube, fourth, fifth, etc…, depending on the index).
2. There is no fraction under the radical sign.
3. There is no radical in the denominator of a fraction.
Quotient Property of Square Roots
4x2
z6
a
a

b
b
is not in simplest form because there is a fraction under the
radical sign. This can be simplified by taking the square root
of the numerator and the denominator.
4x2
4x2 2x

 3
6
6
z
z
z
Simplify
4x2 y
4x2 y

xy
xy
 4x
2 x
Example 2
a)
10
10 6


6
6
6
Multiply numerator by √6 to rationalize the
denominator
Does the radicand have any factors that are
perfect squares? If so, factor them out.
60
6
4 15 2 15 Lastly, cancel out any common factors between


the numerator and denominator.
6
6
15 Completely simplified!

3

b)
3
3
5 35 35
5 3 31 3 15 3 15
3 




9
3
9 3 32 3 32 3 31 3 33
Rewrite the
denominator’s
radicand in
exponential form.
Multiply the
denominator by a
radical with a
factor that will make
the new radicand into
a perfect cube.
When you
write the
denominat
or in
exponential
form, it
3
becomes 3 3
3
Are there
any factors
of 15 that
are perfect
cubes? No.
We are
DONE!