Adding and subtracting radical expressions works like
adding and subtracting expressions

involving variables. Just as we need like terms when combining expressions
involving

variables we need **like radicals** in order to combine radical expressions.

DEFINITION: Two radicals expressions are said to be **
like-radicals** if they have the

same indices and the same radicands.

**EXAMPLE 1**:

**a.** The expressions and
are like-radicals.

**b**. The expressions and
are **not **like radicals since they have
different indices.

**c**. The expressions and
are **not** like radicals since they have
different radicands.

Since only the radicals in **a** are like, we can only
combine (add and subtract) the radicals in **a**.

**EXAMPLE 2**: Add and subtract the pairs of radical
expressions given in **EXAMPLE 1** above.

SOLUTIONS : Since only the radicals in **a** are like, we can only combine
(add or subtract) the

radicals in **a**.

a. ADDITION:

SUBTRACTION:

b. Neither nor
can be simplified since the radicals are not
like

( different indices ).

c. Neither nor
can be simplified since the radicals are not
like

(different radicands).

Sometimes we manipulate the involved radicals so that they
are like, and then combine the

expressions .

**EXAMPLE: **Simplify the following by first obtaining
like-radicals.

SOLUTIONS:

When addition or subtraction is combined with
multiplication, the distributive property is useful.

**EXAMPLE**: Simplify the following.

SOLUTIONS:

When adding or subtracting is combined with division , we
need to rationalize denominators .

Often, rationalizing a denominator can be accomplished by using a cleaver trick
that involves

the **conjugate** of the denominator.

DEFINITION: The **conjugate** of the expression a +b is
the expression a - b.

The conjugate of an expression is a related expression
involving the opposite sign ( +or −).

So the conjugate of expression is the
expression , while the conjugate of

is .

Conjugates are useful when rationalizing denominators since the product of two
conjugates

contains no radicals:

**EXAMPLE:** Simplify

SOLUTION: If we multiply the denominator by its conjugate,
we will have rationalized the

denominator since the denominator will contain no radicals. In order to avoid

changing the expression , we must also multiply the numerator by the conjugate of

the denominator.

**EXAMPLE: **Simplify the following.

SOLUTIONS:

Try these yourself and check your answers.

Simplify the following.

SOLUTIONS: