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.

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

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 :

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 :

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 :

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.

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 :

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.

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…

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…

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

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

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

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

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

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

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…

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…

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 :

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…

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

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.

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 :

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…

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…