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


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.

baseexponent

Like this:

25say 2 to the 5th power


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


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

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 = base2

22say 2 to the 2nd power or twosquared

MOST mathematicians say two squared

22=2 x 2=4


cubed = base3

23say 2 to the 3rd power or twocubed

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= -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:

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

Anynumber(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= 4,638

anynumber1 = the same base “any number”

01 = 0


The exponent1

is

usually

invisible.


Theinvisibleexponent 1

21=2

4,6381= 4,638

anynumber1 = the same base “any number”

01 = 0


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…”

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 5x5x5so

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 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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