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2. Radicals and Rational Exponents
Chapter 10

3. 10 Radicals and Rational Exponents 10.1 Finding Roots
10.3 Simplifying Expressions Containing Square Roots
10.4 Simplifying Expressions Containing Higher Roots
10.5 Adding, Subtracting, and Multiplying Radicals
10.6 Dividing Radicals
Putting it All Together
10.7 Solving Radical Equations
10.8 Complex Numbers

4. 10.5 Adding, Subtracting, and Multiplying Radicals
Add and Subtract Radical Expressions
Examples of Like Radicals
The following are examples of Radicals that are not Like Radicals
We add and subtract like radicals in the same way we add and subtract like terms-
add or subtract the coefficients of the radicals and multiply the result by the radical.
We are using the distributive property when we combine like terms in this way.

5. Perform the operations and simplify.
Example 1
Perform the operations and simplify.
Solution
a) First notice that 9x and 2x are like terms. Therefore, they can be added.
Distributive property
Simplify.
Or, we can say that by just adding the coefficients, 9x + 2x = 11x.
b) First notice that these are like radicals. They have the same index and the same
radicand. Therefore, they can be added.
Distributive property
Simplify.

6. Perform the operations and simplify.
Example 2
Perform the operations and simplify.
Solution
First notice that there are two different types of radicals. Write the like radicals
together.
Commutative property
Distributive property.

7. Simplify Before Adding and Subtracting
Radicals sometimes look as if they cannot be added or subtracted. It is important to check to see if the radicals can be simplified. Sometimes by simplifying the radicals turn out to be like radicals, then we can add or subtract.
Example 3
Perform the operations and simplify.
Solution
Factor
Product Rule.
Simplify radicals.
Multiply.
Add like radicals.

8. Perform the operations and simplify.
Example 4
Perform the operations and simplify.
Solution
Factor
Product Rule.
Simplify radicals.
Multiply.
Subtract like radicals.
Product Rule.
Simplify radicals.
Add like radicals.

9. Multiply a Binomial Containing Radical Expressions by a Monomial
Example 5
Multiply and simplify.
Solution
Factor
Product Rule.
Simplify Radical Rule.
Add Radicals in parenthesis.
Multiply.
Distribute
Product Rule
Factor
Simplify Radical Rule.

10. Example 6 Multiply and simplify. Solution Factor Product Rule.
Distribute.
Multiply
Simplify

11. F o I L Multiply Radical Expressions Using FOIL.
Recall, that in chapter 6 we first multiplied binomials using FOIL (First, Outer, Inner
Last). We can multiply binomials containing radicals in the same way.
Example 6
Multiply and simplify.
F o I L
Solution

12. Example 7
Multiply and simplify.
Solution
F o I L

13. Square a Binomial Containing Radical Expressions
Example 8
Multiply and simplify.
Solution
Multiply
Combine like terms.

14. Example 9
Multiply and simplify.
Solution
Square each term.
Simplify.
When we multiply expressions of the form (a + b) (a – b) containing square
Roots, the radicals are eliminated. This will always be true.
Note