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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.

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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.

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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. 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

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**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 :

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**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 :

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**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 :

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**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.

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**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 :

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**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.

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**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…

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**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…

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**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

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**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

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**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

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**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

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**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

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**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

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**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…

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**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…

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**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 :

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**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…

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**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

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**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.

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**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 :

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**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…

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**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…

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Identify the perfect square in each set. 1. 45, 81, 27, 111 2. 156, 99, 8, 25 3. 256, 84, 12, 10004. 35, 216, 196, 72 Find the Prime Factorization of.

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