roots and powers rational numbers, irrational numbers chapter 4
TRANSCRIPT
![Page 1: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/1.jpg)
ROOTS and
POWERS
Rational numbers, irrational numbers
CHAPTER 4
![Page 2: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/2.jpg)
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}
![Page 3: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/3.jpg)
Natural Numbers
Whole Numbers
Integers
Rational Numbers
Irrational Num
bers
Real NumbersIm
aginary Num
bers
Complex Numbers
![Page 4: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/4.jpg)
1.1.4
![Page 5: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/5.jpg)
Review
RADICALS
![Page 6: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/6.jpg)
Index
Radicand
When the index of the radical is not shown then it is understood to be an index of 2
Radical
𝟑√𝟔𝟒
=
![Page 7: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/7.jpg)
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
![Page 8: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/8.jpg)
Evaluate each radical:
√36
= 0.5
= 6= 2=
= 5
EXAMPLE 2:
![Page 9: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/9.jpg)
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:
![Page 10: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/10.jpg)
4.2 Irrational Numbers
![Page 11: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/11.jpg)
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:
![Page 12: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/12.jpg)
2. Which of these radicals are rational numbers? Which ones are not rational numbers?
How do you know?
WORK WITH YOUR PARTNER
![Page 13: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/13.jpg)
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
![Page 14: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/14.jpg)
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
![Page 15: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/15.jpg)
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
![Page 16: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/16.jpg)
POWER POINT PRACTICE PROBLEMTell whether each number is rational or
irrational. Explain how do you know.
![Page 17: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/17.jpg)
EXAMPLE 2:Use a number line to order these numbers from
least to greatest
Use Calculators!
![Page 18: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/18.jpg)
-2 -1 0 1 2 3 4 5
EXAMPLE 2:Use a number line to order these numbers from
least to greatest
![Page 19: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/19.jpg)
POWERPOINT PRACTICE PROBLEMUse a number line to order these numbers from
least to greatest
![Page 20: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/20.jpg)
HOMEWORKO PAGES: 211 - 212O PROBLEMS: 3 – 6, 9, 15, 20, 18, 19
4.2
![Page 21: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/21.jpg)
4.3 Mixed and Entire Radicals
![Page 22: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/22.jpg)
![Page 23: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/23.jpg)
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 =
![Page 24: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/24.jpg)
MULTIPLICATION PROPERTY of RADICALS
Use Your Calculator to calculate:
What do you notice?
![Page 25: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/25.jpg)
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
![Page 26: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/26.jpg)
Example 1:❑√𝟐𝟒=√𝟒 ·√𝟔
¿𝟐 ·√𝟔¿𝟐√𝟔
![Page 27: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/27.jpg)
Example 2:
𝟑√𝟐𝟒=𝟑√𝟑 ·𝟖¿𝟑√𝟑·𝟑√𝟖¿𝟐𝟑√𝟑
![Page 28: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/28.jpg)
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
![Page 29: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/29.jpg)
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
![Page 30: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/30.jpg)
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
![Page 31: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/31.jpg)
POWERPOINT PRACTICE PROBLEMSimplify each radical.
![Page 32: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/32.jpg)
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
![Page 33: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/33.jpg)
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
![Page 34: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/34.jpg)
Mixed Radical: the product of a number and a
radical
4 6Entire Radical:
the product of one and a radical
72
![Page 35: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/35.jpg)
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 =
![Page 36: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/36.jpg)
𝒏√𝒂𝒃=𝒏√𝒂 ·𝒏√𝒃Writing Mixed Radicals as Entire
Radicals
𝒏√𝒂 ·𝒏√𝒃=𝒏√𝒂𝒃
![Page 37: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/37.jpg)
Write each mixed radical as an entire radical
𝒏√𝒂 ·𝒏√𝒃=𝒏√𝒂𝒃
![Page 38: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/38.jpg)
POWERPOINT PRACTICE PROBLEMWrite each mixed radical as an entire
radical
![Page 39: ROOTS and POWERS Rational numbers, irrational numbers CHAPTER 4](https://reader035.vdocuments.net/reader035/viewer/2022081420/551aad1b5503466b3a8b5e71/html5/thumbnails/39.jpg)
HOMEWORKO PAGES: 218 - 219O PROBLEMS: 4, 5, 7, 11 – 12 (a, d, e,
h, i), 15 – 18, 19, 20
4.3