Rationalize the denominator and multiply with Radicals
Math 51 Worksheet
Rationalize the denominator and multiply with Radicals
Rationalizing is done to remove the radical from the denominator of a fraction. We will consider three cases involving square roots. It will be helpful to remember how to reduce a radical when continuing with these problems.
( )2
Note: Squaring a radical will eliminate the radical. Because x x = x = x this equals x
Case 1: Single radical in the denominator
2y
o
18 y3
So we can eliminate the radical from the denominator by doubling it.
o 2y
But anything you do the bottom of a fraction you must do to the top.
18 y3 18 y 3
o
2y
18 y 3
18 y3 18 y 3
Now multiply across
2 y 18y 3
( )2 18 y 3
then simplify both top and bottom.
2y 92y2y
o
18 y3
23 y y 2y
Reduce the radical on top
18 y3
6y2 2y
o
18 y 3
Reduce the fraction using the terms outside the radical
2y
Done!
3y
Case 2: Entire fraction is in the radical
50 x 3
o
6x4
First, reduce under the radical whenever possible.
25
o
3x
Then, split the radical top and bottom
25
Continue as in case 1 above.
3x
o
25
3x
75x Simplify the radicals 5 3x Done!
3x 3x
9x2
3x
Rationalizing two terms in the denominator requires the conjugate. Definition: Conjugates are the same two terms separated by different signs.
For instance: 5 - 11 and 5 + 11 are conjugates of each other.
Example: Rationalize the denominator
5
o
The conjugate of
2- 3
2 - 3 is
2 + 3 so
5 2 + 3 Multiply 2- 3 2+ 3
5( 2 + 3)
( )( ) o
FOIL the denominator
2- 3 2+ 3
5( 2 + 3)
2 2+ 2 3- 2 3- 3 3
o
5( 2 + 3)
(5
2+
3)
(5
2+
3)
4+ 6- 6- 9
2-3
-1
( ) o Reduce bringing the minus sign to the numberator - 5 2 + 3 Done
Multiply and FOIL with radicals. Example 1: Distribution with a single term
( ) ? 5y 5 - y Use distribution
5y 5 - 5y y
Rules for radicals 5y 5 = 25y and 5y y = 5 y 2
? 25 y - 5 y 2 Simplify the radicals 5 y - y 5 Done!
Example 2: FOIL multiply two terms with two terms
First
Outer Inner Last
( )( ) ? 2x - 3y x - 3y 2x x - 2x 3y - 3y x + 3y 3y
? 2x 2 - 6xy - 3xy + 9 y 2 Answer x 2 - 6xy - 3xy + 3y
Example 3: Two terms squared; double then FOIL
( ) ( )( ) 2
? 3 - w 3 - w 3 - w Must write two terms then FOIL
First Outer Inner Last
? 33-3 w -3 w + w w
combine radicals and simplify
? 9 - 6 w + w Finished because none of the terms are like terms.
Practice Problems
Rationalize and multiply the radicals.
24 x 3
1)
8y
27
2)
6
3
3)
2y
2
4)
6 -3
5+2
5)
2- 3
4
6)
x
( ) 7) 7 8 + 12
8) ( 6 - 3)( 3 + 18)
9) ( 3 - 5)( 3 + 5)
( )2
10) 2x - 7
x 3xy
1)
y 6 y
3)
2y 5) 2 5 + 15 + 4 + 2 3 7) 8 7 + 2 21 9) - 22
Answer Key
3 2
2)
2
4)
2 -
6+6
3
2 x
6)
x
8) 6 3 + 3 2 - 3 - 3 6
10) 2x -14 2x + 49
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