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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