algebraic expressions – rationalizing the denominator when working with radicals in fractions, the...
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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.
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.
xxxx
2
3933
2422
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.
xxxx
2
3933
2422
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
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 :
a
2
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 :
a
2 To rationalize, we will multiply both the numerator and denominator by aa
a
a
a
a
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 1 :
a
2To rationalize, we will multiply both the numerator and denominator by aa
a
a
a
a
22
EXAMPLE 2 :6
2
xx
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 :
a
2To rationalize, we will multiply both the numerator and denominator by aa
a
a
a
a
22
EXAMPLE 2 :6
2
xx
6
62
6
6
6
2
x
xx
x
x
x
x
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.
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 :
x
x
12
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 3 :
x
x
x
x
x
x
x
x
3
2
32
4
34
4
12
4
In some cases, the denominator can be simplified before you rationalize. It will save you steps in the long run.
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 :
x
x
x
x
x
x
x
x
3
2
32
4
34
4
12
4
Now we can rationalize and simplify wherever needed…
3
32
3
32
3
3
3
2 x
x
xx
x
x
x
x
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.
22 bababa
A conjugate is the binomial partner to the expression which completes a difference of squares. Let’s create a table so you can see…
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.
22 bababa
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
9x
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.
22 bababa
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
9x 9x
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.
22 bababa
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
9x 9x53
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.
22 bababa
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
9x 9x53 53
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.
22 bababa
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
9x 9x53 53
ba
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.
22 bababa
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
9x 9x53 53
ba ba
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.
22 bababa
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
9x 9x53 53
ba ba
product
Use the above shortcut or FOIL…
22 ba
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.
22 bababa
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
9x 9x53 53
ba ba
product
81x
22253
ba
Notice how the radical disappears…
22 ba
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 :52
3
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 :52
52
52
3
Multiply top and bottom by the conjugate…
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 : 252
523
52
52
52
3
I like to simplify the denominator first. That way if I can, I can reduce using the integer outside in the numerator
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 : 23
523
252
523
52
52
52
3
No reducing is possible so this is the final answer.
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 :
7
3
b
a
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 :
7
7
7
3
b
b
b
a
Multiply top and bottom by the conjugate…
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 : 7
73
7
7
7
3
b
ba
b
b
b
a
This is your final answer…