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:

25 say 2 to the 5th power

The base(big number on the bottom)=

the repeated factor in a multiplication problem.

base exponent = powerfactor 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.

25

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”

25 say 2 to the 5th power

Ordinal numbers: 1st, 2nd, 3rd, 4th, 5th,…

squared = base2

22 say 2 to the 2nd power or two squared

MOST mathematicians say two squared

22 = 2 x 2 = 4

cubed = base3

23 say 2 to the 3rd power or two cubed

MOST mathematicians say two cubed

23 = 2 x 2 x 2 = 8

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

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.

Common mistake

-24 = (-1)x(x means times) +24 =

-1 x +2 x +2 x +2 x +2 = -16Why?

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 = +16Why? Multiply the numbers: 2 x 2 x 2 x 2 = 16 andthen 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.

20 = 1

4,6380 = 1

Any number(except the number 0)0 = 1

00 = undefined

When the exponent is 1,the answer is the same number as the base number.

21 = 2 4,6381 = 4,638any number1 = the same

base “any number”01 = 0

The exponent 1 is

usually invisible.

The invisible exponent 1

21 = 2 4,6381 = 4,638any number1 = the same base “any number”

01 = 0

2 = 2 4,638 = 4,638any number = the same “any number” as the base

0 = 0The 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 25 = 2 x 2 x 2 x 2 x 2

product = write the long way = answer

“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)

“Write prime factorization using exponents…”

125 = product 5 x 5 x 5 so

125 = power 53 = 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 ExponentsMathematics: Applications and Concepts Course 2:

(blue book) Chapter 1 Lesson 2: Powers and ExponentsPre-Algebra: (green book) Chapter 4 Lesson 2: Powers and ExponentsFor more information on my math class see http://

walsh.edublogs.org