Download Aim: How do we rationalize a denominator?

Aim: How do we rationalize a denominator?
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Factor completely:
2x2 – 50
Aim: Rationalize Denominator
Course: Adv. Alg. & Trig.
Conjugates
2x2 - 50 = 2(x – 5)(x + 5)
conjugates of each other
General Terms a2 – b2 = (a – b)(a + b)
When conjugates are multiplied, the result
is the difference between perfect squares.
What is the conjugate of
(x – 6) (x + 6)
= x2 – 36
(x  5) (x  5) = x2 – 5
(3  7) (3  7) = 9 – 7 = 2
Aim: Rationalize Denominator
Course: Adv. Alg. & Trig.
Rational & Irrational Numbers
Rational Numbers - Any number, integer,
fraction, decimal; positive or negative,
mixed or improper, that can be expressed
as a fraction.
a
y
b
Ex. 2, 1.765, 1/2, 0.33
Every rational number can be expressed as
either a repeating or terminating decimal.
Irrational Numbers - Any and all
numbers that can not be expressed as a
fraction.
Ex.
.439439543957. . .
p = 3.1415926535897932 . . .
2 = 1.4141213562 . . . Course: Adv. Alg. & Trig.
Aim: Rationalize Denominator
Real Number Family
Rational
Numbers
Irrational
Numbers
Integers
Whole Numbers
Counting
Numbers
Counting Numbers
1, 2, 3, 4, 5, . . .
Whole Numbers
0, 1, 2, 3, 4, 5, . .
Integers
. . . -3, -2, -1, 0, 1, 2, 3, . .
Rational Numbers
a/b
Irrational
Numbers
a/b
Aim: Rationalize Denominator
Course: Adv. Alg. & Trig.
Rationalizing a Monomial Denominator
means to remove
the irrational number
from the denominator
10
8
10
3
rational number
irrational number
Multiply fraction by a form of
the identity element 1.
10 
 8 
  10 8  10 8
 8  8  ( 8) 2
8
Simplify the radical, if possible
10 8 10 2 4 10  2 2



8
8
8
5 2
2
Aim: Rationalize Denominator
Course: Adv. Alg. & Trig.
Rationalizing a Denominator
2
4  11 binomial denominator
(of a fraction where the denominator is not
a rational number) means to find a
denominator in which the denominator is
a2 – b2 = (a – b)(a + b)
a rational number.
How can we use the conjugate to rationalize
2
 (4  11)
4  11  (4  11)
Multiply fraction by a form of
the identity element 1.
 2 
4  11 
  8  2 11  8  2 11
4  11 4  11 
16  11
5
Aim: Rationalize Denominator
Course: Adv. Alg. & Trig.
Model Problems
Rationalize the denominator:
3
3
3
 3 18  3 18 
6
6 
3


   



3
3
3
3
4 9
4 9  3   4 27   12 
6
Express
as an equivalent fraction
with a 3  5
rational denominator.
6 
3  5 
 6(3  5)  3(3  5)
3  5 3  5 
4
2
71
Write an equivalent expression for
72
3
 7  1  7  2 7  3 7  2 9  3 7




3
 7  2  7  2
3
 3 7
Aim: Rationalize Denominator
Course: Adv. Alg. & Trig.
Model Problems
Simplify/Rationalize:
6
43 3
1)
2)
8
33 2
3) ( 3  2)  ( 2  8)
4
4)
15  3
5)
6)
7
3 7
5 2 1
2 2 1
Aim: Rationalize Denominator
Course: Adv. Alg. & Trig.
Model Problems
Simplify/Rationalize:
1)
6
8
2)
43 3
33 2
Aim: Rationalize Denominator
Course: Adv. Alg. & Trig.
Model Problems
Simplify/Rationalize:
3) ( 3  2)  ( 2  8)
Aim: Rationalize Denominator
Course: Adv. Alg. & Trig.
Model Problems
Simplify/Rationalize:
4
4)
15  3
Aim: Rationalize Denominator
Course: Adv. Alg. & Trig.
Model Problems
Simplify/Rationalize:
5)
7
3 7
Aim: Rationalize Denominator
Course: Adv. Alg. & Trig.
Model Problems
Simplify/Rationalize:
5 2 1
6)
2 2 1
Aim: Rationalize Denominator
Course: Adv. Alg. & Trig.