The document discusses powers and exponents. It explains that multiplication is a shortcut for repeated addition, and exponents are a shortcut for repeated multiplication. An exponent written as a base number with a little number on top, where the base is the number being multiplied and the exponent tells how many times to multiply the base by itself. Common mistakes in working with exponents are also described.

1. Powers and Exponents

2. Multiplication = short-cut addition When you need to add the same number to itself over and over again, multiplication is a short-cut way to write the addition problem . Instead of adding 2 + 2 + 2 + 2 + 2 = 10 multiply 2 x 5 (and get the same answer) = 10

3. Powers = short-cut multiplication When you need to multiply the same number by itself over and over again, powers are a short-cut way to write the multiplication problem . Instead of multiplying 2 x 2 x 2 x 2 x 2 = 32 Use the power 2 5 (and get the same answer) = 32

4. A power = a number written as a base number with an exponent. base exponent Like this: 2 5 say 2 to the 5th power

5. The base (big number on the bottom) = the repeated factor in a multiplication problem. base exponent = power factor x factor x factor x factor x factor = product 2 x 2 x 2 x 2 x 2 = 32

6. The exponent (little number on the top right of base) = the number of times the base is multiplied by itself. 2 5 2 (1 st time) x 2 (2 nd time) x 2 (3 rd time) x 2 (4 th time) x 2 (5 th time) = 32

7. How to read powers and exponents Normally, say “ base number to the exponent number (expressed as ordinal number) power” 2 5 say 2 to the 5th power Ordinal numbers: 1 st , 2 nd , 3 rd , 4 th , 5 th ,…

8. squared = base 2 2 2 say 2 to the 2nd power or two squared MOST mathematicians say two squared 2 2 = 2 x 2 = 4

9. cubed = base 3 2 3 say 2 to the 3rd power or two cubed MOST mathematicians say two cubed 2 3 = 2 x 2 x 2 = 8

10. Common Mistake 2 5 ≠ (does not equal) 2 x 5 2 5 ≠ (does not equal) 10 2 5 = 2 x 2 x 2 x 2 x 2 = 32

11. Common Mistake - 2 4 ≠ (does not equal) ( - 2 ) 4 With out the parenthesis, positive 2 is multiplied by itself 4 times; then the answer is negative. With the parenthesis, negative 2 is multiplied by itself 4 times; then the answer becomes positive.

12. Common mistake - 2 4 = (- 1 )x (x means times) + 2 4 = - 1 x + 2 x + 2 x + 2 x + 2 = - 16 Why? The 1 and the positive sign are invisible. Anything x 1=anything, so 1 x 2 x 2 x 2 x 2 = 16; and negative x positive = negative

13. Common Mistake ( - 2 ) 4 = - 2 x -2 x -2 x -2 = +16 Why? Multiply the numbers: 2 x 2 x 2 x 2 = 16 and then multiply the signs: 1 st negative x 2 nd negative = positive; that positive x 3 rd negative = negative; that negative x 4 th negative = positive; so answer = positive 16

14. When the exponent is 0 , and the base is any number but 0, the answer is 1 . 2 0 = 1 4,638 0 = 1 Any number (except the number 0) 0 = 1 0 0 = undefined

15. When the exponent is 1 , the answer is the same number as the base number . 2 1 = 2 4,638 1 = 4,638 any number 1 = the same base “any number” 0 1 = 0

16. The exponent 1 is usually invisible .

17. The invisible exponent 1 2 1 = 2 4,638 1 = 4,638 any number 1 = the same base “any number” 0 1 = 0

18. 2 = 2 4,638 = 4,638 any number = the same “any number” as the base 0 = 0 The exponent 1 is here. Can you see it? It’s invisible. Or. It’s understood. The invisible exponent 1

19. “Write a power as a product…” power = write the short-cut way means 2 5 = 2 x 2 x 2 x 2 x 2 product = write the long way = answer

20. “Find the value of the product…” means answer 2 5 = 2 x 2 x 2 x 2 x 2 = 32 power = product = value of the product (and value of the power)

21. “ Write prime factorization using exponents…” 125 = product 5 x 5 x 5 so 125 = power 5 3 = answer using exponents product 5 x 5 x 5 = power 5 3 Same exact answer written two different ways.

22. Congratulations! Now you know how to write a multiplication problem as a product using factors, or as a power using exponents (this can be called exponential form ). You know how to (evaluate) find the value (answer) of a power.

23. Notes for teachers Correlates with Glencoe Mathematics (Florida Edition) texts: Mathematics: Applications and Concepts Course 1: (red book) Chapter 1 Lesson 4 Powers and Exponents Mathematics: Applications and Concepts Course 2: (blue book) Chapter 1 Lesson 2: Powers and Exponents Pre-Algebra: (green book) Chapter 4 Lesson 2: Powers and Exponents For more information on my math class see http:// walsh.edublogs.org