Powers and Exponents: Understanding Multiplication and Shortcuts

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 25 (and get the same answer) = 32

4. A power = a number written as a base number with an exponent. baseexponent Like this: 25say 2 to the 5th power

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

6. Theexponent(little number on the top right of base) = the number of times the base is multiplied by itself. 25 2(1st time) x 2(2nd time) x 2(3rd time) x 2(4th time) x 2(5thtime) = 32

7. How to read powers and exponents Normally, say “base number to the exponent number (expressed as ordinal number) power” 25say2 to the 5th power Ordinal numbers: 1st, 2nd, 3rd, 4th, 5th,…

8. squared = base2 22say 2 to the 2nd power or twosquared MOST mathematicians say two squared 22=2 x 2=4

9. cubed = base3 23say 2 to the 3rd power or twocubed MOST mathematicians say two cubed 23=2 x 2 x 2=8

10. Common Mistake 25 ≠(does not equal)2 x 5 25 ≠(does not equal)10 25 =2 x 2 x 2 x 2 x 2= 32

11. Common Mistake -24 ≠(does not equal)(-2)4 Without 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 -24 = (-1)x(x means times)+24 = -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: 1st negative x 2nd negative = positive; that positive x 3rd negative = negative; that negative x 4th negative = positive; so answer = positive 16

14. When the exponent is 0, and the base is any number but 0, the answer is 1. 20=1 4,6380= 1 Anynumber(except the number 0)0 = 1 00 = undefined

15. When the exponent is 1, the answer is the same number as the base number. 21=2 4,6381= 4,638 anynumber1 = the same base “any number” 01 = 0

16. The exponent1 is usually invisible.

17. Theinvisibleexponent 1 21=2 4,6381= 4,638 anynumber1 = the same base “any number” 01 = 0

18. The invisibleexponent 1 2=2 4,638= 4,638 anynumber = 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.

19. “Write a power as a product…” power = write the short-cut way means 25 = 2 x 2 x 2 x 2 x 2 product = write the long way = answer

20. “Find the value of the product…” means answer 25 = 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 5x5x5so 125 = power 53 = answer using exponents product 5 x 5 x 5 = power 53 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