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**Rational numbers, irrational numbers**

CHAPTER 4 ROOTS and POWERS Rational numbers, irrational numbers

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**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}

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**Complex Numbers Real Numbers Rational Numbers Integers**

Whole Numbers Natural Numbers Irrational Numbers Imaginary Numbers

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1.1.4

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Review RADICALS

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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

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**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

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**Evaluate each radical:**

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

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**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 = …

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4.2 Irrational Numbers

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**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

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**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

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**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

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**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

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**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

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POWER POINT PRACTICE PROBLEM Tell whether each number is rational or irrational. Explain how do you know.

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**EXAMPLE 2: Use a number line to order these numbers from least to greatest **

𝟒 𝟐𝟕 𝟏𝟖 𝟗 Use Calculators! 𝟑 −𝟓 𝟑 𝟏𝟑

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**EXAMPLE 2: Use a number line to order these numbers from least to greatest **

𝟑 −𝟓 𝟒 𝟐𝟕 𝟏𝟖 𝟑 𝟏𝟑 𝟗

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

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

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**4.3 Mixed and Entire Radicals**

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**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

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**MULTIPLICATION PROPERTY of RADICALS**

Use Your Calculator to calculate: What do you notice?

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**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

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Example 1: 𝟐𝟒 = 𝟒 · 𝟔 =𝟐· 𝟔 =𝟐 𝟔

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

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**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

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**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

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**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

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**POWERPOINT PRACTICE PROBLEM Simplify each radical.**

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**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

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**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

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**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

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**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

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**Writing Mixed Radicals as Entire Radicals**

𝒏 𝒂𝒃 = 𝒏 𝒂 · 𝒏 𝒃 𝒏 𝒂 · 𝒏 𝒃 = 𝒏 𝒂𝒃

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**Write each mixed radical as an entire radical**

𝒏 𝒂 · 𝒏 𝒃 = 𝒏 𝒂𝒃

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**POWERPOINT PRACTICE PROBLEM **

Write each mixed radical as an entire radical

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

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