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Algebra I – Roots and Radicals

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    Study Guide Algebra I – Roots and Radicals 1. Introduc Ɵ on to Roots and Radicals Let’s begin with the basics. What Is a Radical? The symbol √ is called a radical sign . It is used to show a square root . For example: • √ 9 = 3 because 3 × 3 = 9 • √ 16 = 4 because 4 × 4 = 16 In simple terms, a square root asks: “What number multiplied by itself gives this result?” What About Cube Roots? If we want to find a cube root , we use a small 3 written above the radical sign: This tells us we are looking for a number that is multiplied by itself three times . For example: • ∛ 8 = 2 because 2 × 2 × 2 = 8 • ∛ 27 = 3 because 3 × 3 × 3 = 27 That small number above the radical sign is called the index . If there is no number written there, it automatically means square root (index 2) .

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    Study Guide What If Two Radicals Are Next to Each Other? If two radical expressions are written next to each other, they are multiplied . For example: The multiplication sign is usually not written , but it is understood. Square Roots of Negative Numbers Now here’s something important: In the real number system , you cannot take the square root of a negative number. For example: There is no real number that you can multiply by itself to get − 4. So we need a new system of numbers. Imaginary Numbers To handle square roots of negative numbers, mathematicians created a new system called imaginary numbers . The key idea is the imaginary unit: This means:

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    Study Guide • √− 1 = i • √− 4 = 2i • √− 9 = 3i Here’s why: Imaginary numbers are simply multiples of i . 2. Simplifying Square Roots Now that you understand what square roots are, let’s learn how to simplify them step by step. The goal of simplifying is to rewrite a radical in its simplest form — meaning no perfect square factors are left inside the radical. Example 1: Basic Square Roots Let’s start with some simple cases. 1. √ 9 Because 3 × 3 = 9. 2. −√ 9 Be careful here!

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    Study Guide The negative sign is outside the radical. So we first find √ 9 = 3, then apply the negative sign. This notation (writing −√ 9 instead of √− 9) is commonly used in textbooks, and we will follow that same style. 3. √ 18 Break 18 into a perfect square times another number: Now split the radical: We simplified it because 9 is a perfect square. Square Roots of Variables Now let’s look at expressions with variables. 4. √ x² If we are told that each variable is nonnegative , then: Why? Because x × x = x². What If the Variable Could Be Positive or Negative?

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    Study Guide If x might be positive or negative, we must use absolute value : Why absolute value? Because a square root is always nonnegative. Absolute value makes sure the answer is never negative. For example: • If x = 5 → √ x² = 5 • If x = − 5 → √ x² = 5 So we write |x| to cover both cases. More Variable Examples e. √ x ⁴ If x is nonnegative: Because: f. √ (x ⁶ y ⁸ ) If each variable is nonnegative:

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    Study Guide If the variables could be positive or negative, we must use absolute value for odd powers: Since y ⁴ is always nonnegative, this is often written simply as: g. √ (25a ⁴ b ⁶ ) If each variable is nonnegative: Break it apart: If the variables could be positive or negative, use absolute value for odd powers: Bigger Example h. √ x ⁷ If x is nonnegative:

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    Study Guide If x could be positive or negative: i. √ (x ⁹ y ⁸ ) If variables are nonnegative: Break into perfect squares: j. √ (16x ⁵ ) If variables are nonnegative:

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    Study Guide 3. Quiz: Simplifying Square Roots 1. Question Which of the following expressions results in an imaginary number? Answer Choices • √ ( - 3)² • √ ( - 3)³ • √ ( - 3) ⁴ Correct Answer : √ ( - 3)³ Why This Is Correct Let’s evaluate each expression: • ( - 3)² = 9 → √ 9 = 3 (real number) • ( - 3)³ = - 27 → √ ( - 27) is the square root of a negative number (imaginary) • ( - 3) ⁴ = 81 → √ 81 = 9 (real number) Only √ ( - 3)³ results in the square root of a negative number, which is imaginary. 2. Question Which of the following correctly states the value of √ 25? Answer Choices • ±5 • - 5

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    Study Guide • +5 Correct Answer : +5 Why This Is Correct The square root symbol ( √ ) represents the principal (positive) square root . Even though both 5 and - 5 squared equal 25, √ 25 is defined as the positive root only. So: √ 25 = 5 3. Question If x could be positive or negative, then which of the following is the best way to express √ (x¹ ⁰ )? Answer Choices • |x ⁵ | • x ⁵ • - x ⁵ Correct Answer : |x ⁵ | Why This Is Correct √ (x¹ ⁰ ) = √ (x ⁵ )² The square root of something squared equals its absolute value when the variable could be positive or negative. So: √ (x¹ ⁰ ) = |x ⁵ | Absolute value ensures the result is never negative.

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    Study Guide 4. Question √ 48 when completely simplified would be: Answer Choices • 2 √ 12 • 4 √ 3 • already simplified Correct Answer : 4 √ 3 Why This Is Correct Break 48 into a perfect square: 48 = 16 × 3 √ 48 = √ 16 × √ 3 = 4 √ 3 Since 3 has no perfect square factors, this is fully simplified. 5. Question Which of the following cannot be simplified? Answer Choices : √ 75 • √ 27 • √ 62 Correct Answer : √ 62

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