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 = 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 2 5 (and get the same answer) = 32

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

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

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

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,…

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

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

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

Common Mistake -2 4 ≠ (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.

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

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

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

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

The exponent 1 is usually invisible.

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

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

“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

“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)

“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.

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