1. 1 Lesson 2.5.1 Powers of Integers

2. 2 Lesson 2.5.1 Powers of Integers California Standards: Number Sense 1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers. 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 how to write repeated multiplications in a shorter form. Key words: power base exponent factor

3. 3 Lesson 2.5.1 Powers of Integers A power is just the product that you get when you repeatedly multiply a number by itself, like 2 2, or 3 3 3. Repeated multiplication expressions can be very long. So there’s a special system you can use for writing out powers in a shorter way — that’s what this Lesson is about.

4. 4 A Power is a Repeated Multiplication Lesson 2.5.1 Powers of Integers A power is a product that results from repeatedly multiplying a number by itself. For example: 2 2 = 4, or “two to the second power.” 2 2 2 = 8, or “two to the third power.” 2 2 2 2 = 16, or “two to the fourth power.” So 4, 8, and 16 are all powers of 2.

5. 5 You Can Write a Power as a Base and an Exponent Lesson 2.5.1 Powers of Integers If every time you used a repeated multiplication you wrote it out in full, it would make your work very complicated. So there’s a shorter way to write them. For example: 2 2 2 2 = 2 4 2 is the base — it’s the number that’s being multiplied This is the exponent — it tells you how many times the base number is a factor in the multiplication expression.

6. 6 Lesson 2.5.1 Powers of Integers The expression 10 10 can be written in this form too — the base is 10 and since 10 occurs twice, the exponent is 2. 10 2 Base Exponent You can rewrite any repeated multiplication in this form. So any number, x, to the n th power can be written as: xnxn Base Exponent

7. 7 Example 1 Solution follows… Lesson 2.5.1 Powers of Integers Write the expression: 3 3 3 3 3 in base and exponent form. Solution The number that is being multiplied is 3. So the base is 3. So 3 3 3 3 3 = 3 5. 3 occurs as a factor five times in the multiplication expression. So the exponent is 5.

8. 8 Lesson 2.5.1 Powers of Integers If a number has an exponent of 1 then it occurs only once in the expanded multiplication expression. So any number to the power 1 is just the number itself. For example: 5 1 = 5, 137 1 = 137, x 1 = x.

9. 9 Guided Practice Solution follows… Lesson 2.5.1 Powers of Integers Write each of the expressions in Exercises 1–8 as a power in base and exponent form. 1. 8 82. 2 2 2 3. 7 7 7 7 74. 5 5. 9 9 9 96. 4 4 4 4 4 4 4 7. –5 –58. –8 –8 –8 –8 –8 8282 2323 7575 5151 9494 4747 (–5) 2 (–8) 5

10. 10 Evaluate a Power By Doing the Multiplication Lesson 2.5.1 Powers of Integers Evaluating a power means working out its value. Just write it out in its expanded form — then treat it as any other multiplication calculation.

11. 11 Example 2 Solution follows… Lesson 2.5.1 Powers of Integers Evaluate 5 4. Solution 5 4 means “four copies of the number five multiplied together.” 5 4 = 5 5 5 5 5 4 = 625 Write the expression in expanded form Then do the multiplication

12. 12 Example 3 Solution follows… Lesson 2.5.1 Powers of Integers Evaluate (–2) 2. Solution (–2) 2 means “two copies of negative two multiplied together.” (–2) 2 = –2 –2 (–2) 2 = 4 Write the expression in expanded form Then do the multiplication

13. 13 Guided Practice Solution follows… Lesson 2.5.1 Powers of Integers Evaluate the exponential expressions in Exercises 9–16. 9. 10 2 10. 5 3 11. 7 1 12. 3 6 13. 47 1 14. (–15) 1 15. (–3) 2 16. (–4) 3 10 10 = 1005 5 5 = 125 73 3 3 3 3 3 = 729 47–15 –3 –3 = 9–4 –4 –4 = –64

14. 14 Independent Practice Solution follows… Lesson 2.5.1 Powers of Integers Write each of the expressions in Exercises 1–6 in base and exponent form. 1. 4 4 4 2. 9 9 3. 8 4. 5 5 5 5 5 5 5. –4 –4 –46. –3 –3 –3 –3 4343 9292 8181 5656 (–4) 3 (–3) 4

15. 15 Independent Practice Solution follows… Lesson 2.5.1 Powers of Integers 7. Kiera and 11 of her friends are handing out fliers for a school fund-raiser. Each person hands out fliers to 12 people. How many people receive a flier? 12 12 = 12 2 = 144

16. 16 Independent Practice Solution follows… Lesson 2.5.1 Powers of Integers Evaluate the exponential expressions in Exercises 8–13. 8. 15 2 9. 4 3 10. 8 1 11. 1 8 12. (–5) 3 13. (–5) 4 22564 81 –125625

17. 17 Independent Practice Solution follows… Lesson 2.5.1 Powers of Integers 14. A single yeast cell is placed on a nutrient medium. This cell will divide into two cells after one hour. These two cells will then divide to form four cells after another hour. The process continues indefinitely. a) How many yeast cells will be present after 1 hour, 2 hours, and 6 hours? b) Write exponential expressions with two as the base to describe the number of yeast cells that will be present after 1 hour, 2 hours, and 6 hours. c) How many hours will it take for the yeast population to reach 256? 1 hour = 2 cells, 2 hours = 4 cells, 6 hours = 64 cells 1 hour = 2 1 cells, 2 hours = 2 2 cells, 6 hours = 2 6 cells 8 hours

18. 18 Round Up Lesson 2.5.1 Powers of Integers If you need to use a repeated multiplication, it’s useful to have a shorter way of writing it. That’s why bases and exponents come in really handy when you’re writing out powers of numbers. You’ll see lots of powers used in expressions, equations, and formulas. For example, the formula for the area of a circle is  r 2 where r is the radius. So it’s important you know what they mean.