Unit 1B

Limits

Lesson 2: Algebraic Limits

Skill Builder

Removing Radicals

Removing Radicals from the Denominator: Rationalizing the Denominator
Simplified radical expressions do not have radicals in the denominator. The process of removing the radicals is called rationalizing the denominator. To rationalize the denominator, you will apply the rule «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mroot»«msup»«mi»x«/mi»«mi»n«/mi»«/msup»«mi»n«/mi»«/mroot»«mo»=«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math».

Denominators with Square Roots
For rational expressions with square root denominators, multiply the denominator by itself in order to remove the radical sign. You must also remember to multiply the numerator by the same value so that you do not change the expression! This is similar to saying you have multiplied by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math», since «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfrac»«msqrt»«mi»a«/mi»«/msqrt»«msqrt»«mi»a«/mi»«/msqrt»«/mfrac»«mo»=«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math». This process is demonstrated in Example 1.

Simplify «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfrac»«mn»3«/mn»«msqrt»«mn»5«/mn»«/msqrt»«/mfrac»«/mstyle»«/math».

To eliminate «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msqrt»«mn»5«/mn»«/msqrt»«/mstyle»«/math» from the denominator, you must multiply it by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msqrt»«mn»5«/mn»«/msqrt»«/mstyle»«/math». If you were just to multiply the denominator by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msqrt»«mn»5«/mn»«/msqrt»«/mstyle»«/math», you would change the expression. To overcome this issue, multiply the numerator by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msqrt»«mn»5«/mn»«/msqrt»«/mstyle»«/math» as well.

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mrow mathcolor=¨#191919¨»«mfrac»«mn»3«/mn»«msqrt»«mn»5«/mn»«/msqrt»«/mfrac»«mo»§#215;«/mo»«mn»1«/mn»«/mrow»«/mtd»«mtd»«mo mathcolor=¨#191919¨»=«/mo»«/mtd»«mtd»«mrow mathcolor=¨#191919¨»«mfrac»«mn»3«/mn»«msqrt»«mn»5«/mn»«/msqrt»«/mfrac»«mo»§#8729;«/mo»«mfrac»«msqrt»«mn»5«/mn»«/msqrt»«msqrt»«mn»5«/mn»«/msqrt»«/mfrac»«/mrow»«/mtd»«/mtr»«mtr»«mtd»«mrow mathcolor=¨#191919¨/»«/mtd»«mtd»«mo mathcolor=¨#191919¨»=«/mo»«/mtd»«mtd»«mfrac mathcolor=¨#191919¨»«mrow»«mn»3«/mn»«msqrt»«mn»5«/mn»«/msqrt»«/mrow»«msqrt»«msup»«mn»5«/mn»«mn»2«/mn»«/msup»«/msqrt»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mrow mathcolor=¨#191919¨/»«/mtd»«mtd»«mo mathcolor=¨#191919¨»=«/mo»«/mtd»«mtd»«mfrac mathcolor=¨#191919¨»«mrow»«mn»3«/mn»«msqrt»«mn»5«/mn»«/msqrt»«/mrow»«mn»5«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»

Notice that the exponent on the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»5«/mn»«/mstyle»«/math», in the second line of the solution, is the same as the index of the radical.

Simplify «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfrac»«mrow»«mn»6«/mn»«msqrt»«mn»2«/mn»«/msqrt»«/mrow»«mrow»«mn»5«/mn»«msqrt»«mn»3«/mn»«mi»b«/mi»«/msqrt»«/mrow»«/mfrac»«/mstyle»«/math».

Step 1:
Identify just the radical in the denominator.

In this case, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»5«/mn»«msqrt»«mn»3«/mn»«mi»b«/mi»«/msqrt»«/mrow»«/mstyle»«/math» is the radical. Note that you do not multiply by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»5«/mn»«msqrt»«mn»3«/mn»«mi»b«/mi»«/msqrt»«/mrow»«/mstyle»«/math» because «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»5«/mn»«/mstyle»«/math» is not a radical.

Step 2:
Multiply both numerator and denominator by that radical.

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mfrac»«mrow»«mn»6«/mn»«msqrt»«mn»2«/mn»«/msqrt»«/mrow»«mrow»«mn»5«/mn»«msqrt»«mn»3«/mn»«mi»b«/mi»«/msqrt»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»6«/mn»«msqrt»«mn»2«/mn»«/msqrt»«/mrow»«mrow»«mn»5«/mn»«msqrt»«mn»3«/mn»«mi»b«/mi»«/msqrt»«/mrow»«/mfrac»«mo»§#8729;«/mo»«mfrac»«msqrt»«mn»3«/mn»«mi»b«/mi»«/msqrt»«msqrt»«mn»3«/mn»«mi»b«/mi»«/msqrt»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mrow mathcolor=¨#000000¨/»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»6«/mn»«msqrt»«mn»2«/mn»«mo»§#8729;«/mo»«mn»3«/mn»«mi»b«/mi»«/msqrt»«/mrow»«mrow»«mn»5«/mn»«msqrt»«msup»«mn»3«/mn»«mn»2«/mn»«/msup»«msup»«mi»b«/mi»«mn»2«/mn»«/msup»«/msqrt»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mrow mathcolor=¨#000000¨/»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»6«/mn»«msqrt»«mn»6«/mn»«mi»b«/mi»«/msqrt»«/mrow»«mrow»«mn»5«/mn»«mo»§#8729;«/mo»«mn»3«/mn»«mi»b«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mrow mathcolor=¨#000000¨/»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»2«/mn»«msqrt»«mn»6«/mn»«mi»b«/mi»«/msqrt»«/mrow»«mrow»«mn»5«/mn»«mi»b«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»

Watch for the simplification that can occur after rationalizing the denominator, such as removing a factor of three from both the numerator and the denominator in this example.


Conjugates

The binomials «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mi»a«/mi»«mo»+«/mo»«mi»b«/mi»«/mrow»«/mfenced»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mi»a«/mi»«mo»-«/mo»«mi»b«/mi»«/mrow»«/mfenced»«/mstyle»«/math» are conjugates, and their product is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msup»«mi»a«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«msup»«mi»b«/mi»«mn»2«/mn»«/msup»«/mrow»«/mstyle»«/math».

Simplify «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfrac»«mn»3«/mn»«mrow»«msqrt»«mn»3«/mn»«/msqrt»«mo»§#8722;«/mo»«msqrt»«mn»6«/mn»«/msqrt»«/mrow»«/mfrac»«/mstyle»«/math».

Step 1:
Determine the conjugate of the denominator.

The conjugate of the denominator is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msqrt»«mn»3«/mn»«/msqrt»«mo»+«/mo»«msqrt»«mn»6«/mn»«/msqrt»«/mrow»«/mstyle»«/math».

Step 2:
Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, and then simplify.

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mfrac»«mn»3«/mn»«mrow»«msqrt»«mn»3«/mn»«/msqrt»«mo»§#8722;«/mo»«msqrt»«mn»6«/mn»«/msqrt»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»3«/mn»«mfenced»«mrow»«msqrt»«mn»3«/mn»«/msqrt»«mo»§#8722;«/mo»«msqrt»«mn»6«/mn»«/msqrt»«/mrow»«/mfenced»«/mfrac»«mo»§#8226;«/mo»«mfrac»«mfenced»«mrow»«msqrt»«mn»3«/mn»«/msqrt»«mo»+«/mo»«msqrt»«mn»6«/mn»«/msqrt»«/mrow»«/mfenced»«mfenced»«mrow»«msqrt»«mn»3«/mn»«/msqrt»«mo»+«/mo»«msqrt»«mn»6«/mn»«/msqrt»«/mrow»«/mfenced»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mrow mathcolor=¨#000000¨/»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»3«/mn»«mfenced»«mrow»«msqrt»«mn»3«/mn»«/msqrt»«mo»+«/mo»«msqrt»«mn»6«/mn»«/msqrt»«/mrow»«/mfenced»«/mrow»«mrow»«mfenced»«mrow»«msqrt»«mn»3«/mn»«/msqrt»«mo»§#8722;«/mo»«msqrt»«mn»6«/mn»«/msqrt»«/mrow»«/mfenced»«mfenced»«mrow»«msqrt»«mn»3«/mn»«/msqrt»«mo»+«/mo»«msqrt»«mn»6«/mn»«/msqrt»«/mrow»«/mfenced»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mrow mathcolor=¨#000000¨/»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»3«/mn»«mfenced»«mrow»«msqrt»«mn»3«/mn»«/msqrt»«mo»+«/mo»«msqrt»«mn»6«/mn»«/msqrt»«/mrow»«/mfenced»«/mrow»«mrow»«msqrt»«msup»«mn»3«/mn»«mn»2«/mn»«/msup»«/msqrt»«menclose notation=¨updiagonalstrike¨»«mo»+«/mo»«msqrt»«mn»3«/mn»«/msqrt»«mo»§#8226;«/mo»«msqrt»«mn»6«/mn»«/msqrt»«/menclose»«menclose notation=¨updiagonalstrike¨»«mo»§#8722;«/mo»«msqrt»«mn»3«/mn»«/msqrt»«mo»§#8226;«/mo»«msqrt»«mn»6«/mn»«/msqrt»«/menclose»«mo»§#8722;«/mo»«msqrt»«msup»«mn»6«/mn»«mn»2«/mn»«/msup»«/msqrt»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mrow mathcolor=¨#000000¨/»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»3«/mn»«mfenced»«mrow»«msqrt»«mn»3«/mn»«/msqrt»«mo»+«/mo»«msqrt»«mn»6«/mn»«/msqrt»«/mrow»«/mfenced»«/mrow»«mrow»«msqrt»«msup»«mn»3«/mn»«mn»2«/mn»«/msup»«/msqrt»«mo»§#8722;«/mo»«msqrt»«msup»«mn»6«/mn»«mn»2«/mn»«/msup»«/msqrt»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mrow mathcolor=¨#000000¨/»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»3«/mn»«mfenced»«mrow»«msqrt»«mn»3«/mn»«/msqrt»«mo»+«/mo»«msqrt»«mn»6«/mn»«/msqrt»«/mrow»«/mfenced»«/mrow»«mrow»«mn»3«/mn»«mo»§#8722;«/mo»«mn»6«/mn»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mrow mathcolor=¨#000000¨/»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»3«/mn»«mfenced»«mrow»«msqrt»«mn»3«/mn»«/msqrt»«mo»+«/mo»«msqrt»«mn»6«/mn»«/msqrt»«/mrow»«/mfenced»«/mrow»«mrow»«mo»§#8722;«/mo»«mn»3«/mn»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mrow mathcolor=¨#000000¨/»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mfenced»«mrow»«msqrt»«mn»3«/mn»«/msqrt»«mo»+«/mo»«msqrt»«mn»6«/mn»«/msqrt»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mrow mathcolor=¨#000000¨/»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«msqrt»«mn»3«/mn»«/msqrt»«mo»§#8722;«/mo»«msqrt»«mn»6«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Watch the video below for more information on rationalizing.

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