Rational numbers, irrational numbers CHAPTER 4 ROOTS and POWERS Rational numbers, irrational numbers

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: 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}

Complex Numbers Real Numbers Rational Numbers Integers Whole Numbers Natural Numbers Irrational Numbers Imaginary Numbers

1.1.4

Review RADICALS

Index Radical 𝟑 𝟔𝟒 Radicand When the index of the radical is not shown then it is understood to be an index of 2 𝟔𝟒 = 𝟐 𝟔𝟒

EXAMPLE 1: Give 4 examples of radicals Use a different radicand and index for each radical Explain the meaning of the index of each radical

Evaluate each radical: EXAMPLE 2: Evaluate each radical: 36 = 6 3 8 = 2 3 27 125 = 𝟑 𝟓 4 625 = 5 3 0.125 = 0.5

EXAMPLE 3: 16 = 4 3 −27 = -3 25/16 = 5/4 2 = 1.4141… A whole number Choose values of n and x so that 𝒏 𝒙 is: A whole number A negative integer A rational number An approximate decimal 16 = 4 3 −27 = -3 25/16 = 5/4 2 = 1.4141…

4.2 Irrational Numbers

These are rational numbers: These are NOT rational numbers: WORK WITH YOUR PARTNER 1. How are radicals that are rational numbers different from radicals that are not rational numbers? Rational Numbers: These are rational numbers: These are NOT rational numbers: 9 64 1 3 0.25 2 0.5 3 9 0.24 3 8 100 5 −32 4 12

4 5 64 81 WORK WITH YOUR PARTNER 1.44 3 −27 5 How do you know? 2. Which of these radicals are rational numbers? Which ones are not rational numbers? How do you know? 4 5 64 81 1.44 3 −27 5

25 100 64 81 RATIONAL NUMBERS 16 3 −27 36 Can be written in the form Radicals that are square roots of perfect squares, cube roots of perfect cubes etc.. They have decimal representation which terminate or repeats 25 100 64 81 16 3 −27 36

4 5 π 64 81 IRATIONAL NUMBERS 14 5 π 2 Can not be written in the form They are non-repeating and non-terminating decimals 4 5 64 81 14 5 π π 2

EXAMPLE 1: Tell whether each number is rational or irrational EXAMPLE 1: Tell whether each number is rational or irrational. Explain how do you know. 𝟑 𝟖 𝟐𝟕 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 𝟏𝟒 𝟎.𝟓 Rational, because 0.5 terminates. π Irrational, because π is not a repeating decimal and does not terminates

POWER POINT PRACTICE PROBLEM Tell 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 𝟒 𝟐𝟕 𝟏𝟖 𝟗 Use Calculators! 𝟑 −𝟓 𝟑 𝟏𝟑

EXAMPLE 2: Use a number line to order these numbers from least to greatest 𝟑 −𝟓 𝟒 𝟐𝟕 𝟏𝟖 -2 -1 0 1 2 3 4 5 𝟑 𝟏𝟑 𝟗

POWERPOINT PRACTICE PROBLEM Use a number line to order these numbers from least to greatest

HOMEWORK PAGES: 211 - 212 PROBLEMS: 3 – 6, 9, 15, 20, 18, 19 4.2

4.3 Mixed and Entire Radicals

𝟑 𝟔𝟒 𝟔𝟒 = 𝟐 𝟔𝟒 Review of Radicals Index Radical Radicand When the index of the radical is not shown then it is understood to be an index of 2. 𝟔𝟒 = 𝟐 𝟔𝟒

MULTIPLICATION PROPERTY of RADICALS Use Your Calculator to calculate: What do you notice?

MULTIPLICATION PROPERTY of RADICALS 𝒏 𝒂𝒃 = 𝒏 𝒂 · 𝒏 𝒃 where n is a natural number, and a and b are real numbers 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

Example 1: 𝟐𝟒 = 𝟒 · 𝟔 =𝟐· 𝟔 =𝟐 𝟔

Example 2: 𝟑 𝟐𝟒 = 𝟑 𝟑·𝟖 = 𝟑 𝟑 · 𝟑 𝟖 =𝟐 𝟑 𝟑

Write each radical as a product of prime factors, then simplify. 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

Write each radical as a product of prime factors, then simplify. 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

Write each radical as a product of prime factors, then simplify. 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 PROBLEM Simplify each radical.

Writing Radicals in Simplest Form 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, 200 Since 1, 4, 16, 25, 100, and 400 are perfect squares, we can simplify √400 in several ways: 𝟐𝟎𝟎 = 𝟏·𝟐𝟎𝟎 =𝟏 𝟐𝟎𝟎 𝟐𝟎𝟎 = 𝟒·𝟓𝟎 = 𝟒 · 𝟓𝟎 = 𝟐 𝟓𝟎 𝟐𝟎𝟎 = 𝟐𝟓·𝟖 = 𝟐𝟓 · 𝟖 =𝟓 𝟖 𝟐𝟎𝟎 = 𝟏𝟎𝟎·𝟐 = 𝟏𝟎𝟎 · 𝟐 =𝟏𝟎 𝟐

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 = 𝟒·𝟓𝟎 = 𝟒 · 𝟓𝟎 = 𝟐 𝟓𝟎 𝟐𝟎𝟎 = 𝟐𝟓·𝟖 = 𝟐𝟓 · 𝟖 =𝟓 𝟖 𝟐𝟎𝟎 = 𝟏𝟎𝟎·𝟐 = 𝟏𝟎𝟎 · 𝟐 =𝟏𝟎 𝟐

the product of a number and a radical Mixed Radical: the product of a number and a radical Entire Radical: the product of one and a radical

Writing Mixed Radicals as Entire Radicals Any number can be written as the square root of its square! 2 = 𝟐•𝟐 45 = 𝟒𝟓•𝟒𝟓 100 = 𝟏𝟎𝟎•𝟏𝟎𝟎 Any number can be also written as the cube root of its cube, or the fourth root of its perfect fourth! 2 = 𝟓 𝟐•𝟐•𝟐•𝟐•𝟐 45 = 𝟒 𝟒𝟓•𝟒𝟓•𝟒𝟓•𝟒𝟓

Writing Mixed Radicals as Entire Radicals 𝒏 𝒂𝒃 = 𝒏 𝒂 · 𝒏 𝒃 𝒏 𝒂 · 𝒏 𝒃 = 𝒏 𝒂𝒃

Write each mixed radical as an entire radical 𝒏 𝒂 · 𝒏 𝒃 = 𝒏 𝒂𝒃

POWERPOINT PRACTICE PROBLEM Write each mixed radical as an entire radical

HOMEWORK PAGES: 218 - 219 PROBLEMS: 4, 5, 7, 11 – 12 (a, d, e, h, i), 15 – 18, 19, 20 4.3