1.
Powers and Exponents

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

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

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

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

6.
<ul><li>The exponent (little number on the top right of base) = the number of times the base is multiplied by itself. </li></ul><ul><li>2 5 </li></ul><ul><li>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 </li></ul>

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

8.
squared = base 2 <ul><li>2 2 say 2 to the 2nd power or two squared </li></ul><ul><li>MOST mathematicians say two squared </li></ul><ul><li>2 2 = 2 x 2 = 4 </li></ul>

9.
cubed = base 3 <ul><li>2 3 say 2 to the 3rd power or two cubed </li></ul><ul><li>MOST mathematicians say two cubed </li></ul><ul><li>2 3 = 2 x 2 x 2 = 8 </li></ul>

10.
Common Mistake <ul><li>2 5 ≠ (does not equal) 2 x 5 </li></ul><ul><li>2 5 ≠ (does not equal) 10 </li></ul><ul><li>2 5 = 2 x 2 x 2 x 2 x 2 = 32 </li></ul>

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

12.
Common mistake <ul><li>- 2 4 = (- 1 )x (x means times) + 2 4 = </li></ul><ul><li>- 1 x + 2 x + 2 x + 2 x + 2 = - 16 </li></ul>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 <ul><li>( - 2 ) 4 = - 2 x -2 x -2 x -2 = +16 </li></ul>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 , <ul><li>and the base is any number but 0, the answer is 1 . </li></ul><ul><li>2 0 = 1 </li></ul><ul><li>4,638 0 = 1 </li></ul><ul><li>Any number (except the number 0) 0 = 1 </li></ul><ul><li>0 0 = undefined </li></ul>

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

16.
<ul><li>The exponent 1 </li></ul><ul><li>is </li></ul><ul><li>usually </li></ul><ul><li>invisible . </li></ul>

17.
The invisible exponent 1 <ul><li>2 1 = 2 </li></ul><ul><li>4,638 1 = 4,638 </li></ul><ul><li>any number 1 = the same base “any number” </li></ul><ul><li>0 1 = 0 </li></ul>

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

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

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

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

22.
Congratulations! <ul><li>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 ). </li></ul><ul><li>You know how to (evaluate) find the value (answer) of a power. </li></ul>

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