Powers and exponents
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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

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Powers and Exponents

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Powers and exponents

Powers and Exponents


Multiplication short cut addition

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

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


Powers and exponents

A power =

a number written as

a base number with an exponent.

baseexponent

Like this:

25say 2 to the 5th power


Powers and exponents

The base(big number on the bottom)=

the repeatedfactor in a multiplication problem.

baseexponent = power

factor x factor x factor x factor x factor = product

2 x 2 x 2 x 2 x 2 = 32


Powers and exponents

Theexponent(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(5thtime) = 32


How to read powers and exponents

How to read powers and exponents

Normally, say “base number to the exponent number (expressed as ordinal number) power”

25say2 to the 5th power

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


Squared base 2

squared = base2

22say 2 to the 2nd power or twosquared

MOST mathematicians say two squared

22=2 x 2=4


Cubed base 3

cubed = base3

23say 2 to the 3rd power or twocubed

MOST mathematicians say two cubed

23=2 x 2 x 2=8


Common mistake

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 mistake1

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 mistake2

Common mistake

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

-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 mistake3

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:

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

When the exponent is 0,

and the base is any number but 0, the answer is 1.

20=1

4,6380= 1

Anynumber(except the number 0)0 = 1

00 = undefined


When the exponent is 1

When the exponent is 1,

the answer is the same number as the base number.

21=2

4,6381= 4,638

anynumber1 = the same base “any number”

01 = 0


Powers and exponents

The exponent1

is

usually

invisible.


The invisible exponent 1

Theinvisibleexponent 1

21=2

4,6381= 4,638

anynumber1 = the same base “any number”

01 = 0


The invisible exponent 11

The invisibleexponent 1

2=2

4,638= 4,638

anynumber = 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

“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

“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

“Write prime factorization using exponents…”

125 = product 5x5x5so

125 = power 53 = answer using exponents

product 5 x 5 x 5 = power 53

Same exact answer written two different ways.


Congratulations

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

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


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