Download Section 2.5/2.6 – Irrational Numbers/Radicals 1 Section 2.5/2.6

Section 2.5/2.6
Irrational Numbers/Radicals
An irrational number is a number that cannot be written as a fraction and whose
decimal representation is a nonterminating, nonrepeating decimal.
Examples of irrational numbers:
• Pi = π
=
•
Circumference of a circle
= 3.14159265...
Diameter of a circle
Euler’s number = e = 2.7182818284590...
Given a nonnegative number x, in
x the
is the radical symbol and the x is the
radicand. The results of such numbers may be rational or irrational. Given y x the “y”
is called the radical’s coefficient.
The principal square root of
gives the radicand.
x is the positive number that when multiplied to itself, it
A perfect square is any number that is the square of a natural number.
Natural Numbers:
1
2
3
4
5…
Square of the natural numbers:
(perfect squares)
1
4
9
16
25…
Product Rule for Square Roots
Quotient Rule for Square Roots
a
a
=
b
b
ab = a b
Example 1: Simplify.
a.
81
b.
100
c.
36
121
d.
63
e.
124
f.
108
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Section 2.5/2.6 – Irrational Numbers/Radicals
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Rational or Irrational?
In simplifying radicals, if a radical remains then the number is irrational.
Example 2: Classify each of the following numbers as rational or irrational.
a.
32
b.
225
c.
45
d.
126
e.
18
8
f.
240
147
Section 2.5/2.6 – Irrational Numbers/Radicals
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We use the same rules:
ab = a b and
a
a
to multiply or divide.
=
b
b
Example 3: Multiply , then simplify.
a.
3 ⋅ 12
b.
5 ⋅ 10
c.
8 ⋅ 20
d.
98
2
e.
160
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Section 2.5/2.6 – Irrational Numbers/Radicals
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Rationalizing the Denominator
If a fraction has a radical in the denominator (irrational), then it’s not in simplest form.
So we “rationalize the denominator”.
To rationalize the denominator, multiply by a unit fraction that uses the radical in the
denominator then simplify.
Example 4: Simplify (or rationalize the denominator).
1
a.
5
b.
12
6
c.
2
3
d.
e.
7
2 7
50
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Section 2.5/2.6 – Irrational Numbers/Radicals
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Addition and Subtraction of Irrational Numbers
We can add and subtract irrational numbers only if they have like radicals.
Example 5: Add or subtract.
a. 5 6 + 6
b. 10 11 − 4 11
c. 2 5 − 3 20
d.
3 − 2 12 + 3 48
Note:
x + y ≠ x + y and
x− y ≠ x − y
Section 2.5/2.6 – Irrational Numbers/Radicals
5
Other Radical Forms
The k th root of a nonnegative number x denoted
k
k
x is the number n such that
x =n
means n k = x .
For example, 3 1000 = 10 because 103 = 1000 !
Example 6: Simplify.
3
a.
3
27
b.
4
16
c.
16
2
3
Fractional Exponents
If m and n are integers and n > 0, then
n
m
n
=
x m x=
( ).
x
1
m
n
Example 7: Simplify.
a.
( 36 )
1
c.
( 32 )
−2
e.
3
y12
g.
3
x9
y15
b. ( 8 )
2
5
Section 2.5/2.6 – Irrational Numbers/Radicals
5
3
d.
x6
f.
x 4 y 8 z10
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