ROOTS and
POWERS
Rational numbers, irrational numbers
CHAPTER 4
THE REAL NUMBER SYSTEM
Natural Numbers: N = { 1, 2, 3, …}Whole Numbers: W = { 0, 1, 2 , 3, ...}Integers: I = {….. -3, -2, -1, 0, 1, 2, 3, ...}
Rational Numbers: Q a
b| a,b I ,b 0
Irrational Numbers: Q = {non-terminating, non-repeating decimals} π, e ,√2 , √ 3 ...Real Numbers: R = {all rational and irrational}
Imaginary Numbers: i = {square roots of negative numbers}
Complex Numbers: C = { real and imaginary numbers}
Natural Numbers
Whole Numbers
Integers
Rational Numbers
Irrational Num
bers
Real NumbersIm
aginary Num
bers
Complex Numbers
1.1.4
Review
RADICALS
Index
Radicand
When the index of the radical is not shown then it is understood to be an index of 2
Radical
𝟑√𝟔𝟒
=
EXAMPLE 1:
a)Give 4 examples of radicals
b)Use a different radicand and index for each radical
c) Explain the meaning of the index of each radical
Evaluate each radical:
√36
= 0.5
= 6= 2=
= 5
EXAMPLE 2:
Choose values of n and x so that is:
a) A whole number
b) A negative integer
c) A rational number
d) An approximate decimal
= 4
= 5/4
= 1.4141…
= -3
EXAMPLE 3:
4.2 Irrational Numbers
WORK WITH YOUR PARTNER
1. How are radicals that are rational numbers different from radicals that are not rational numbers?
Rational Numbers: Q a
b| a,b I ,b 0
These are rational numbers: These are NOT rational numbers:
2. Which of these radicals are rational numbers? Which ones are not rational numbers?
How do you know?
WORK WITH YOUR PARTNER
RATIONAL NUMBERSa. Can be written in the formb. Radicals that are square roots of perfect squares,
cube roots of perfect cubes etc..c. They have decimal representation which
terminate or repeats
Q a
b| a,b I ,b 0
IRATIONAL NUMBERS
a. Can not be written in the formb. They are non-repeating and non-terminating
decimals
Q a
b| a,b I ,b 0
EXAMPLE 1: Tell whether each number is rational or irrational. Explain how do you
know.
Rational, because 8/27 is a perfect cube. Also, 2/3 or 0.666… is a repeating decimal.
Irrational, because 14 is not a perfect square. Also, √14 is NOT a repeating decimal and DOES NOT
terminate
Rational, because 0.5 terminates.
Irrational, because π is not a repeating decimal and does not terminates
POWER POINT PRACTICE PROBLEMTell whether each number is rational or
irrational. Explain how do you know.
EXAMPLE 2:Use a number line to order these numbers from
least to greatest
Use Calculators!
-2 -1 0 1 2 3 4 5
EXAMPLE 2:Use a number line to order these numbers from
least to greatest
POWERPOINT PRACTICE PROBLEMUse a number line to order these numbers from
least to greatest
HOMEWORKO PAGES: 211 - 212O PROBLEMS: 3 – 6, 9, 15, 20, 18, 19
4.2
4.3 Mixed and Entire Radicals
Index
Radicand
Review of Radicals
When the index of the radical is not shown then it isunderstood to be an index of 2.
Radical
𝟑√𝟔𝟒 =
MULTIPLICATION PROPERTY of RADICALS
Use Your Calculator to calculate:
What do you notice?
𝒏√𝒂𝒃=𝒏√𝒂 ·𝒏√𝒃
WE USE THIS PROPERTY TO: Simplify square roots and cube roots
that are not perfect squares or perfect cubes, but have factors that are perfect squares/cubes
MULTIPLICATION PROPERTY of RADICALS
where n is a natural number, and a and b are real numbers
Example 1:❑√𝟐𝟒=√𝟒 ·√𝟔
¿𝟐 ·√𝟔¿𝟐√𝟔
Example 2:
𝟑√𝟐𝟒=𝟑√𝟑 ·𝟖¿𝟑√𝟑·𝟑√𝟖¿𝟐𝟑√𝟑
Simplify each radical.
Write each radical as a product of prime factors, then simplify.
Since √80 is a square root. Look for factors that appear twice
Simplify each radical.
Write each radical as a product of prime factors, then simplify.
Since 144 ∛ is a cube root. Look for factors that appear three times
Simplify each radical.
Write each radical as a product of prime factors, then simplify.
Since 162 ∜ is a fourth root. Look for factors that appear four times
POWERPOINT PRACTICE PROBLEMSimplify each radical.
Some numbers such as 200 have more than one perfect square factor:
For example, the factors of 200 are:1, 2 ,4, 5, 8, 10, 20, 25, 40, 50, 100,
200Since 1, 4, 16, 25, 100, and 400 are perfect squares, we can simplify √400 in several ways:
❑√𝟐𝟎𝟎=√𝟏 ·𝟐𝟎𝟎=𝟏√𝟐𝟎𝟎❑√𝟐𝟎𝟎=√𝟐𝟓 ·𝟖=√𝟐𝟓 ·√𝟖=𝟓√𝟖❑√𝟐𝟎𝟎=√𝟏𝟎𝟎 ·𝟐=√𝟏𝟎𝟎 ·√𝟐=𝟏𝟎 √𝟐
Writing Radicals in Simplest Form
Writing Radicals in Simplest Form
10√2 is in simplest form because the radical contains no perfect
square factors other than 1
❑√𝟐𝟎𝟎=√𝟏 ·𝟐𝟎𝟎=𝟏√𝟐𝟎𝟎❑√𝟐𝟎𝟎=√𝟐𝟓 ·𝟖=√𝟐𝟓 ·√𝟖=𝟓√𝟖❑√𝟐𝟎𝟎=√𝟏𝟎𝟎 ·𝟐=√𝟏𝟎𝟎 ·√𝟐=𝟏𝟎 √𝟐
Mixed Radical: the product of a number and a
radical
4 6Entire Radical:
the product of one and a radical
72
Writing Mixed Radicals as Entire RadicalsAny number can be written as the square
root of its square!
2 = 45 = 100 =
Any number can be also written as the cube root of its cube, or the fourth root of
its perfect fourth!2 =
45 =
𝒏√𝒂𝒃=𝒏√𝒂 ·𝒏√𝒃Writing Mixed Radicals as Entire
Radicals
𝒏√𝒂 ·𝒏√𝒃=𝒏√𝒂𝒃
Write each mixed radical as an entire radical
𝒏√𝒂 ·𝒏√𝒃=𝒏√𝒂𝒃
POWERPOINT PRACTICE PROBLEMWrite each mixed radical as an entire
radical
HOMEWORKO PAGES: 218 - 219O PROBLEMS: 4, 5, 7, 11 – 12 (a, d, e,
h, i), 15 – 18, 19, 20
4.3
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