Negative and Zero Exponents Lesson 5.2.1

Negative and Zero Exponents Lesson 5.2.1 Negative and Zero Exponents California Standards: Number Sense 2.1 Understand negative whole-number exponents. Multiply and divide expressions using exponents with a common base. Algebra and Functions 2.1 Interpret positive whole-number powers as repeated multiplication and negative whole-number powers as repeated division or multiplication by the multiplicative inverse. Simplify and evaluate expressions that include exponents. What it means for you: You’ll learn what zero and negative powers mean, and simplify expressions involving them. Key words: base exponent power

Negative and Zero Exponents Lesson 5.2.1 Negative and Zero Exponents Up to now you’ve worked with only positive whole-number exponents. These show the number of times a base is multiplied. As you’ve seen, they follow certain rules and patterns. 83 1212 916 3256 84 122 20 100 4–3 2–6 17–2 6–7 4–10 2–256 The effects of negative and zero exponents are trickier to picture. But you can make sense of them because they follow the same rules and patterns as positive exponents.

Negative and Zero Exponents Lesson 5.2.1 Negative and Zero Exponents Any Number Raised to the Power 0 is 1 Any number that has an exponent of 0 is equal to 1. So, 20 = 1, 30 = 1, 100 = 1, = 1. 1 2 For any number a ¹ 0, a0 = 1 You can show this using the division of powers rule.

Negative and Zero Exponents Lesson 5.2.1 Negative and Zero Exponents If you start with 1000, and keep dividing by 10, you get this pattern: 1000 = 103 100 = 102 10 = 101 1 = 100 Now divide by 10: 103 ÷ 101 = 10(3 – 1) = 102 Now divide by 10: 102 ÷ 101 = 10(2 – 1) = 101 Now divide by 10: 101 ÷ 101 = 10(1 – 1) = 100 1000 = 103 100 = 102 10 = 101 1 = 100 Now divide by 10: 103 ÷ 101 = 10(3 – 1) = 102 Now divide by 10: 102 ÷ 101 = 10(2 – 1) = 101 Now divide by 10: 101 ÷ 101 = 10(1 – 1) = 100 The most important row is the second to last one. When you divide 10 by 10, you have 101 ÷ 101 = 10(1 – 1) = 100. You also know that 10 divided by 10 is 1. So you can see that 100 = 1.

61 ÷ 61 = 6(1 – 1) = 60, and 6 divided by 6 is 1. So 60 = 1. Lesson 5.2.1 Negative and Zero Exponents This pattern works for any base. For instance, 61 ÷ 61 = 6(1 – 1) = 60, and 6 divided by 6 is 1. So 60 = 1. You can use the fact that any number to the power 0 is 1 to simplify expressions.

Negative and Zero Exponents Lesson 5.2.1 Negative and Zero Exponents Example 1 Simplify 34 × 30. Leave your answer in base and exponent form. Solution 34 × 30 = 34 × 1 = 34 You can use the multiplication of powers rule to show this is right: 34 × 30 = 3(4 + 0) = 34 Add the exponents of the powers You can see that being multiplied by 30 didn’t change 34. Solution follows…

Negative and Zero Exponents Lesson 5.2.1 Negative and Zero Exponents Guided Practice Evaluate the following. 1. 40 2. x0 (x ¹ 0) 3. 110 + 120 4. (7 + 6)0 5. 43 ÷ 43 6. y2 ÷ y2 (y ¹ 0) 7. 32 × 30 8. 24 × 20 9. a8 ÷ a0 (a ¹ 0) 1 1 2 1 1 1 32 or 9 24 or 16 a8 Solution follows…

Negative and Zero Exponents Lesson 5.2.1 Negative and Zero Exponents You Can Justify Negative Exponents in the Same Way By continuing the pattern of powers shown below you can begin to understand the meaning of negative exponents. 1000 = 103 100 = 102 10 = 101 1 = 100 Now divide by 10: 103 ÷ 101 = 10(3 – 1) = 102 Now divide by 10: 102 ÷ 101 = 10(2 – 1) = 101 Now divide by 10: 101 ÷ 101 = 10(1 – 1) = 100

Negative and Zero Exponents Lesson 5.2.1 Negative and Zero Exponents Carry on dividing each power of 10 by 10: 100 = 102 10 = 101 1 = 100 Now divide by 10: 102 ÷ 101 = 10(2 – 1) = 101 Now divide by 10: 101 ÷ 101 = 10(1 – 1) = 100 Now divide by 10: 100 ÷ 101 = 10(0 – 1) = 10–1 1 10 = 10–1 Now divide by 10: 10–1 ÷ 101 = 10(–1 – 1) = 10–2 1 100 = 10–2 Now divide by 10: 10–2 ÷ 101 = 10(–2 – 1) = 10–3 1 1000 = 10–3

Negative and Zero Exponents Lesson 5.2.1 Negative and Zero Exponents Look at the last rows, shown in red, to see the pattern: 1 10 = 10–1 Now divide by 10: 10–1 ÷ 101 = 10(–1 – 1) = 10–2 1 100 = 10–2 Now divide by 10: 10–2 ÷ 101 = 10(–2 – 1) = 10–3 1 1000 = 10–3 One-thousandth, which is , can be rewritten as = 10–3. 1 1000 103 One-hundredth, which is , can be rewritten as = 10–2. 1 100 102 One-tenth, which is , can be rewritten as = 10–1. 1 10 101

Negative and Zero Exponents Lesson 5.2.1 Negative and Zero Exponents This works with any number, not just with 10. For example: 60 = 1 60 ÷ 61 = 6–1 and 1 ÷ 6 = , so 6–1 = . 1 6 6–1 ÷ 61 = 6–2 and ÷ 6 = • = = , so 6–2 = . 1 6 62 36 This pattern illustrates the general definition for negative exponents. For any number a ¹ 0, a–n = 1 an

Negative and Zero Exponents Lesson 5.2.1 Negative and Zero Exponents Example 2 Rewrite 5–3 without a negative exponent. Solution 1 53 5–3 = Using the definition of negative exponents = 1 125 Solution follows…

Negative and Zero Exponents Lesson 5.2.1 Negative and Zero Exponents Example 3 Rewrite using a negative exponent. 1 75 Solution 1 75 = 7–5 Using the definition of negative exponents Solution follows…

Negative and Zero Exponents Lesson 5.2.1 Negative and Zero Exponents Guided Practice Rewrite each of the following without a negative exponent. 10. 7–3 11. 5–m 12. x–2 (x ¹ 0) 1 73 1 5m 1 x2 Solution follows…

Negative and Zero Exponents Lesson 5.2.1 Negative and Zero Exponents Guided Practice Rewrite each of the following using a negative exponent. 13. 14. 15. (q ¹ 0) 1 33 3–3 1 64 6–4 1 q × q × q q–3 Solution follows…

Negative and Zero Exponents Lesson 5.2.1 Negative and Zero Exponents Independent Practice Evaluate the expressions in Exercises 1–3. 1. 87020 2. g0 (g ¹ 0) 3. 20 – 30 1 1 Solution follows…

Negative and Zero Exponents Lesson 5.2.1 Negative and Zero Exponents Independent Practice Write the expressions in Exercises 4–6 without negative exponents. 4. 45–1 5. x–6 (x ¹ 0) 6. y–3 – z–3 (y ¹ 0, z ¹ 0) 1 45 1 x6 1 y3 z3 – Solution follows…

Negative and Zero Exponents Lesson 5.2.1 Negative and Zero Exponents Independent Practice Write the expressions in Exercises 7–9 using negative exponents. 7. 8. (r ¹ 0) 9. (p + q ¹ 0) 1 82 8–2 1 r6 r–6 1 (p + q)v (p + q)–v Solution follows…

Negative and Zero Exponents Lesson 5.2.1 Negative and Zero Exponents Independent Practice In Exercises 10–12, simplify the expression given. 10. 54 × 50 11. c5 × c0 (c ¹ 0) 12. f 3 ÷ f 0 (f ¹ 0) 54 c5 f 3 Solution follows…

Negative and Zero Exponents Lesson 5.2.1 Negative and Zero Exponents Independent Practice 13. The number of bacteria in a Petri dish doubles every hour. The numbers of bacteria after each hour are 1, 2, 4, 8, 16, ... Rewrite these numbers as powers of 2. 20, 21, 22, 23, 24 1 2 4 8 14. Rewrite the numbers 1, , , and as powers of 2. 20, 2–1, 2–2, and 2–3 Solution follows…

Negative and Zero Exponents Lesson 5.2.1 Negative and Zero Exponents Round Up So remember — any number (except 0) to the power of 0 is equal to 1. This is useful when you’re simplifying expressions and equations. Later in this Section, you’ll see how negative powers are used in scientific notation for writing very small numbers efficiently.