Download 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.
( x)
2
Note: Squaring a radical will eliminate the radical. Because
x⋅ x =
=x
⇐ this equals x
Case 1: Single radical in the denominator
o
2y
⇐ So we can eliminate the radical from the denominator by doubling it.
18 y 3
o
2y
18 y 3
o
2y
18 y 3
o
⋅
⋅
18 y 3
18 y 3
2y 9⋅ 2y2 y
18 y
o
18 y 3
⇐ Now multiply across ⇒
2 y 18 y 3
⇐ then simplify both top and bottom.
( 18 y )
2
3
⇐ Reduce the radical on top ⇒
3
6y2 2y
18 y
⇐ But anything you do the bottom of a fraction you must do to the top.
2 ⋅3⋅ y ⋅ y 2y
18 y 3
⇐ Reduce the fraction using the terms outside the radical ⇒
3
2y
Done!
3y
Case 2: Entire fraction is in the radical
o
50 x 3
6x 4
⇐ First, reduce under the radical whenever possible.
o
25
3x
⇐ Then, split the radical top and bottom ⇒
25
o
3x
⋅
3x
3x
⇒
75 x
9x 2
⇐ Simplify the radicals ⇒
5 3x
3x
25
3x
Continue as in case 1 above.
Done!
Rationalizing two terms in the denominator requires the conjugate.
Definition: Conjugates are the same two terms separated by different signs.
5 − 11 and
For instance:
5 + 11 are conjugates of each other.
Example: Rationalize the denominator
5
⇒ The conjugate of
o
2− 3
(
5 2+ 3
o
(
2− 3
)(
2+ 3
(
5 2+ 3
o
)
2 − 3 is
)
⇒
4+ 6− 6− 9
(
5 2+ 3
2−3
)
⇒
(
(
5y
(
5− y
)
← Use distribution ⇒
Rules for radicals ⇒
25 y − 5 y 2
•
5 y ⋅ 5 = 25 y
(
⇒ −5 2 + 3
)
5y ⋅ 5 − 5y ⋅ y
5y ⋅ y = 5y2
and
← Simplify the radicals ⇒
5 y − y 5 Done!
Example 2: FOIL multiply two terms with two terms
•
•
(
2x − 3 y
)(
x − 3y
)
First
⇒
Outer
Inner
Last
2x x − 2x 3y − 3 y x + 3 y 3y
2 x 2 − 6 xy − 3 xy + 9 y 2
Answer
x 2 − 6 xy − 3xy + 3 y
Example 3: Two terms squared; double then FOIL
(3 − w )
2
•
⇒
(3 − w )(3 − w )
First Outer Inner
2+ 3
⇐ Multiply
)
)
Multiply and FOIL with radicals.
Example 1: Distribution with a single term
•
2+ 3
2 2+ 2 3− 2 3− 3 3
5 2+ 3
−1
Reduce bringing the minus sign to the numberator
o
2− 3
⋅
5 2+ 3
⇒ FOIL the denominator ⇒
)
5
2 + 3 so ⇒
⇐ Must write two terms then FOIL
Last
•
3⋅3 − 3 w − 3 w + w w
⇐ combine radicals and simplify
•
9 − 6 w + w Finished because none of the terms are like terms.
Done ☺
Practice Problems
Rationalize and multiply the radicals.
24 x 3
8y
1)
3
3)
5)
9)
5+2
(
7 8 + 12
3 −5
2
6 −3
4
x
6)
2− 3
(
6
4)
2y
7)
27
2)
)(
)
3 +5
8)
)
(
6− 3
(
10)
)(
3 + 18
)
)
2
2x − 7
Answer Key
1)
x 3 xy
y
2)
3)
6y
2y
4)
5)
2 5 + 15 + 4 + 2 3
6)
3 2
2
−
2 6 +6
3
2 x
x
7)
8 7 + 2 21
8)
6 3 +3 2 −3−3 6
9)
− 22
10)
2 x − 14 2 x + 49