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

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 2 5 (and get the same answer) = 32

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

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 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. 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

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

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 ,…

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

2 2 say 2 to the 2nd power or two squared

MOST mathematicians say two squared

2 2 = 2 x 2 = 4

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

2 3 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 5 2 5 ≠ (does not equal) 10 2 5 = 2 x 2 x 2 x 2 x 2 = 32

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

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.

- 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.

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

- 2 4 = (- 1 )x (x means times) + 2 4 =

- 1 x + 2 x + 2 x + 2 x + 2 = - 16

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

( - 2 ) 4 = - 2 x -2 x -2 x -2 = +16

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

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

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 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 exponent 1

is

usually

invisible .

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

2 1 = 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

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 means 2 5 = 2 x 2 x 2 x 2 x 2 product = write the long way = answer

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)

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.

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.

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

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