1. Algebraic Expressions – Rationalizing the Denominator
When working with radicals in fractions, the denominator can not be left as a radical. You must rationalize the denominator using this rule…a radical multiplied by itself equals what is under the radical.

2. Algebraic Expressions – Rationalizing the Denominator
When working with radicals in fractions, the denominator can not be left as a radical. You must rationalize the denominator using this rule…a radical multiplied by itself equals what is under the radical.

3. Algebraic Expressions – Rationalizing the Denominator
When working with radicals in fractions, the denominator can not be left as a radical. You must rationalize the denominator using this rule…a radical multiplied by itself equals what is under the radical.
There are two types of problems we need to consider :
- ones that the denominator contains one term under a radical
- ones where there are multiple terms in the denominator, where one or more terms is under a radical

4. Algebraic Expressions – Rationalizing the Denominator
There are two types of problems we need to consider :
- ones that the denominator contains one term under a radical
- ones where there are multiple terms in the denominator, where one or more terms is under a radical
EXAMPLE 1 :

5. Algebraic Expressions – Rationalizing the Denominator
There are two types of problems we need to consider :
- ones that the denominator contains one term under a radical
- ones where there are multiple terms in the denominator, where one or more terms is under a radical
To rationalize, we will multiply both the numerator and denominator by
EXAMPLE 1 :

6. Algebraic Expressions – Rationalizing the Denominator
There are two types of problems we need to consider :
- ones that the denominator contains one term under a radical
- ones where there are multiple terms in the denominator, where one or more terms is under a radical
To rationalize, we will multiply both the numerator and denominator by
EXAMPLE 1 :
EXAMPLE 2 :

7. Algebraic Expressions – Rationalizing the Denominator
There are two types of problems we need to consider :
- ones that the denominator contains one term under a radical
- ones where there are multiple terms in the denominator, where one or more terms is under a radical
To rationalize, we will multiply both the numerator and denominator by
EXAMPLE 1 :
EXAMPLE 2 :
As long as the square root covers the entire denominator, it is considered one term. You rationalize by multiplying numerator and denominator by the original denominator.

8. Algebraic Expressions – Rationalizing the Denominator
There are two types of problems we need to consider :
- ones that the denominator contains one term under a radical
- ones where there are multiple terms in the denominator, where one or more terms is under a radical
EXAMPLE 3 :

9. Algebraic Expressions – Rationalizing the Denominator
There are two types of problems we need to consider :
- ones that the denominator contains one term under a radical
- ones where there are multiple terms in the denominator, where one or more terms is under a radical
EXAMPLE 3 :
In some cases, the denominator can be simplified before you rationalize. It will save you steps in the long run.

10. Algebraic Expressions – Rationalizing the Denominator
There are two types of problems we need to consider :
- ones that the denominator contains one term under a radical
- ones where there are multiple terms in the denominator, where one or more terms is under a radical
EXAMPLE 3 :
Now we can rationalize and simplify wherever needed…

11. Algebraic Expressions – Rationalizing the Denominator
There are two types of problems we need to consider :
- ones that the denominator contains one term under a radical
- ones where there are multiple terms in the denominator, where one or more terms is under a radical
When more than one term appears in the denominator and at least one term is a radical, we will rationalize using a conjugate, which is based on the difference of squares concept.
A conjugate is the binomial partner to the expression which completes a difference of squares. Let’s create a table so you can see…

12. Algebraic Expressions – Rationalizing the Denominator
There are two types of problems we need to consider :
- ones that the denominator contains one term under a radical
- ones where there are multiple terms in the denominator, where one or more terms is under a radical
When more than one term appears in the denominator and at least one term is a radical, we will rationalize using a conjugate, which is based on the difference of squares concept.
A conjugate is the binomial partner to the expression which completes a difference of squares. Let’s create a table so you can see…
binomial
conjugate

13. Algebraic Expressions – Rationalizing the Denominator
There are two types of problems we need to consider :
- ones that the denominator contains one term under a radical
- ones where there are multiple terms in the denominator, where one or more terms is under a radical
When more than one term appears in the denominator and at least one term is a radical, we will rationalize using a conjugate, which is based on the difference of squares concept.
A conjugate is the binomial partner to the expression which completes a difference of squares. Let’s create a table so you can see…
binomial
conjugate

14. Algebraic Expressions – Rationalizing the Denominator
There are two types of problems we need to consider :
- ones that the denominator contains one term under a radical
- ones where there are multiple terms in the denominator, where one or more terms is under a radical
When more than one term appears in the denominator and at least one term is a radical, we will rationalize using a conjugate, which is based on the difference of squares concept.
A conjugate is the binomial partner to the expression which completes a difference of squares. Let’s create a table so you can see…
binomial
conjugate

15. Algebraic Expressions – Rationalizing the Denominator
There are two types of problems we need to consider :
- ones that the denominator contains one term under a radical
- ones where there are multiple terms in the denominator, where one or more terms is under a radical
When more than one term appears in the denominator and at least one term is a radical, we will rationalize using a conjugate, which is based on the difference of squares concept.
A conjugate is the binomial partner to the expression which completes a difference of squares. Let’s create a table so you can see…
binomial
conjugate

16. Algebraic Expressions – Rationalizing the Denominator
There are two types of problems we need to consider :
- ones that the denominator contains one term under a radical
- ones where there are multiple terms in the denominator, where one or more terms is under a radical
When more than one term appears in the denominator and at least one term is a radical, we will rationalize using a conjugate, which is based on the difference of squares concept.
A conjugate is the binomial partner to the expression which completes a difference of squares. Let’s create a table so you can see…
binomial
conjugate

17. Algebraic Expressions – Rationalizing the Denominator
There are two types of problems we need to consider :
- ones that the denominator contains one term under a radical
- ones where there are multiple terms in the denominator, where one or more terms is under a radical
When more than one term appears in the denominator and at least one term is a radical, we will rationalize using a conjugate, which is based on the difference of squares concept.
A conjugate is the binomial partner to the expression which completes a difference of squares. Let’s create a table so you can see…
binomial
conjugate

18. Algebraic Expressions – Rationalizing the Denominator
There are two types of problems we need to consider :
- ones that the denominator contains one term under a radical
- ones where there are multiple terms in the denominator, where one or more terms is under a radical
When more than one term appears in the denominator and at least one term is a radical, we will rationalize using a conjugate, which is based on the difference of squares concept.
A conjugate is the binomial partner to the expression which completes a difference of squares. Let’s create a table so you can see…
binomial
conjugate
product
Use the above shortcut or FOIL…

19. Algebraic Expressions – Rationalizing the Denominator
There are two types of problems we need to consider :
- ones that the denominator contains one term under a radical
- ones where there are multiple terms in the denominator, where one or more terms is under a radical
When more than one term appears in the denominator and at least one term is a radical, we will rationalize using a conjugate, which is based on the difference of squares concept.
A conjugate is the binomial partner to the expression which completes a difference of squares. Let’s create a table so you can see…
binomial
conjugate
product
Notice how the radical disappears…

20. Algebraic Expressions – Rationalizing the Denominator
There are two types of problems we need to consider :
- ones that the denominator contains one term under a radical
- ones where there are multiple terms in the denominator, where one or more terms is under a radical
EXAMPLE 4 :

21. Algebraic Expressions – Rationalizing the Denominator
There are two types of problems we need to consider :
- ones that the denominator contains one term under a radical
- ones where there are multiple terms in the denominator, where one or more terms is under a radical
EXAMPLE 4 :
Multiply top and bottom by the conjugate…

22. Algebraic Expressions – Rationalizing the Denominator
There are two types of problems we need to consider :
- ones that the denominator contains one term under a radical
- ones where there are multiple terms in the denominator, where one or more terms is under a radical
EXAMPLE 4 :
I like to simplify the denominator first. That way if I can, I can reduce using the integer outside in the numerator

23. Algebraic Expressions – Rationalizing the Denominator
There are two types of problems we need to consider :
- ones that the denominator contains one term under a radical
- ones where there are multiple terms in the denominator, where one or more terms is under a radical
EXAMPLE 4 :
No reducing is possible so this is the final answer.

24. Algebraic Expressions – Rationalizing the Denominator
There are two types of problems we need to consider :
- ones that the denominator contains one term under a radical
- ones where there are multiple terms in the denominator, where one or more terms is under a radical
EXAMPLE 5 :

25. Algebraic Expressions – Rationalizing the Denominator
There are two types of problems we need to consider :
- ones that the denominator contains one term under a radical
- ones where there are multiple terms in the denominator, where one or more terms is under a radical
EXAMPLE 5 :
Multiply top and bottom by the conjugate…

26. Algebraic Expressions – Rationalizing the Denominator
There are two types of problems we need to consider :
- ones that the denominator contains one term under a radical
- ones where there are multiple terms in the denominator, where one or more terms is under a radical
EXAMPLE 5 :
This is your final answer…