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Powers and Exponents 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 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 A power = a number written as a base number with an exponent. base exponent Like this: 5 2 say 2 to the 5th power base(big number on the bottom)= the repeated factor in a multiplication problem. The base exponent = power factor x factor x factor x factor x factor = product 2 x2 x2 x 2 x2 = 32 exponent (little number on the top right of base) = the number of times the base is multiplied by itself. 5 2 The 2(1st time) x 2(2nd time) x 2(3rd time) x 2(4th time) x 2(5th time) = 32 How to read powers and exponents Normally, say “base number to the exponent number (expressed as ordinal number) power” 5 2 say 2 to the 5th power Ordinal numbers: 1st, 2nd, 3rd, 4th, 5th,… squared = base2 2 2 say 2 to the 2nd power or two squared MOST mathematicians say two squared 2 2 =2x2=4 cubed = base3 3 2 say 2 to the 3rd power or two cubed MOST mathematicians say two cubed 3 2 =2x2x2=8 Common Mistake 5 2 ≠ 2x5 5 2 ≠ 10 5 2 =2 x 2 x 2 x 2 x 2= 32 (does not equal) (does not equal) Common Mistake 4 4 -2 ≠(does not equal)(-2) 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. Common mistake 4 -2 = (-1)x (x means times) -1 x +2 x +2 x +2 x +2 4 +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 Common Mistake 4 (-2) = - 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 When the exponent is 0, and the base is any number but 0, the answer is 1. 0 2 =1 0 4,638 = 1 0 Any number(except the number 0) = 1 0 0 = undefined When the exponent is 1, the answer is the same number as the base number. 1 2 =2 4,6381 = 4,638 any number1 = the same base “any number” 1 0 =0 The exponent 1 is usually invisible. The invisible exponent 1 1 2 =2 1 4,638 = 4,638 1 any number = the same base “any number” 1 0 =0 The invisible exponent 1 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. “Write a power as a product…” power = write the short-cut way 5 = means 2 2x2x2x2x2 product = write the long way = answer “Find the value of the product…” means answer 5 2 = 2 x 2 x 2 x 2 x 2 = 32 power = product = value of the product (and value of the power) “Write prime factorization using exponents…” 125 = product 5 x 5 x 5 so 3 125 = power 5 = answer using exponents product 5 x 5 x 5 = power 53 Same exact answer written two different ways. 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. 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