aim: rationalize denominator course: adv. alg. & trig. aim: how do we rationalize a denominator?...
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Aim: Rationalize Denominator Course: Adv. Alg. & Trig.
Aim: How do we rationalize a denominator?
Do Now: Factor completely:
2x2 – 50
Aim: Rationalize Denominator Course: Adv. Alg. & Trig.
General Terms
Conjugates
conjugates of each other
(a – b)(a + b)a2 – b2 =
(x + 6)
(x 5)
What is the conjugate of (x – 6)
(x 5)
(3 7)
(3 7)
When conjugates are multiplied, the result is the difference between perfect squares.
2x2 - 50 = 2(x – 5)(x + 5)
= x2 – 36
= x2 – 5
= 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.
b
ay
Every rational number can be expressed as either a repeating or terminating decimal.
Ex. 2, 1.765, 1/2, 0.33
Irrational Numbers - Any and all
numbers that can not be expressed as a fraction.
Ex. .439439543957. . . = 3.1415926535897932 . . . = 1.4141213562 . . .
2
Aim: Rationalize Denominator Course: Adv. Alg. & Trig.
Rational Numbers
Rational Numbers a/b
Integers
Integers . . . -3, -2, -1, 0, 1, 2, 3, . .
Whole Numbers
Whole Numbers 0, 1, 2, 3, 4, 5, . .
Irrational Numbers
Irrational Numbers a/b
Counting Numbers
Counting Numbers 1, 2, 3, 4, 5, . . .
Real Number Family
Aim: Rationalize Denominator Course: Adv. Alg. & Trig.
Rationalizing a Monomial Denominator
10
8
8
8
10 8
( 8)2
10 8
8
Multiply fraction by a form of the identity element 1.
10
8
Simplify the radical, if possible
10 8
810 2 4
810 2 2
8
5 2
2
means to removethe irrational number from the denominator
10
3 rational number
irrational number
Aim: Rationalize Denominator Course: Adv. Alg. & Trig.
Rationalizing a Denominator
(of a fraction where the denominator is nota rational number) means to find a denominator in which the denominator is a rational number.
How can we use the conjugate to rationalize
2
4 11
(a – b)(a + b)a2 – b2 =
(4 11)
(4 11)
2
4 11
4 11
4 11
8 2 11
16 11
8 2 11
5
Multiply fraction by a form of the identity element 1.
2
4 11 binomial denominator
Aim: Rationalize Denominator Course: Adv. Alg. & Trig.
Model Problems
63
4 93
Rationalize the denominator:
6
3 5Express as an equivalent fractionwith a rational denominator.
6
3 5
Write an equivalent expression for
7 1
7 2
63
4 93
33
33
183
4 273
183
12
3 5
3 5
6(3 5)
4
3(3 5)
2
7 1
7 2
7 2
7 2
7 3 7 2
3
9 3 7
3
3
3 7
Aim: Rationalize Denominator Course: Adv. Alg. & Trig.
Model Problems
Simplify/Rationalize:3
3
6 4 31) 2)
8 3 2
3) ( 3 2) ( 2 8)
44)
15 3
75)
3 7
5 2 16)
2 2 1
Aim: Rationalize Denominator Course: Adv. Alg. & Trig.
Model Problems
Simplify/Rationalize:3
3
6 4 31) 2)
8 3 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:
44)
15 3
Aim: Rationalize Denominator Course: Adv. Alg. & Trig.
Model Problems
Simplify/Rationalize:
75)
3 7
Aim: Rationalize Denominator Course: Adv. Alg. & Trig.
Model Problems
Simplify/Rationalize:
5 2 16)
2 2 1