A b n a n b n a m a m n remarks a radical is in

a
b
=
n
√
a
n
√
b
• n
√
a m = a
m
n
Remarks. A radical is in simplest form if all the following condi-
tions are satisfied:
• The radicand is positive.
• The radicand of a radical of index n has no factor which is a
perfect n th power.
• There are no fractions under the radical sign.
• There is no radical in the denominator of a rational expression.
• The index of the radical is the smallest possible.
Operations Involving Radical Expressions
Addition and Subtraction of Radical Expressions
Radicals with the same index and radicand can be added or sub-
tracted.
Multiplication and Division of Radical Expressions
If radicals have the same index, we use
n
√
a · n
√
b = n
√
ab and
n
√
a
n
√
b
=
n
r
a
b
If radicals have different indices, we first make their indices the same
by finding the LCM of all the indices.
Rationalization of Radical Expressions
To rationalize the denominator,
• If the denominator is a single radical expression, multiply both
the numerator and denominator by an expression that will
make the radicand of the denominator a perfect power of the
index.
• If the denominator consists of two or more radical expres-
sions, use the special products: ( x - y )( x + y ) = x
2
- y
2
,
( x + y )( x
2
- xy + y
2
) = x
3
+ y
3
, and ( x - y )( x
2
+ xy + y
2
) = x
3
- y
3
.