Transcript Powers and Exponents

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:
5
2
say
2 to the 5th power
base(big number on the bottom)=
the repeated factor in a
multiplication
problem.
The
base exponent = power
factor x factor x factor x factor x factor = product
2
x2
x2
x 2
x2
= 32
exponent (little number on the
top right of base) = the number
of times the base is
multiplied by itself.
5
2
The
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”
5
2
say
2 to the 5th power
Ordinal numbers: 1st, 2nd, 3rd, 4th, 5th,…
squared = base2
2
2
say
2 to the 2nd power or two squared
MOST mathematicians say two squared
2
2
=2x2=4
cubed = base3
3
2
say
2 to the 3rd power or two cubed
MOST mathematicians say two cubed
3
2
=2x2x2=8
Common Mistake
5
2 ≠
2x5
5
2 ≠
10
5
2 =2 x 2 x 2 x 2 x 2= 32
(does not equal)
(does not equal)
Common Mistake
4
4
-2 ≠(does not equal)(-2)
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
4
-2 =
(-1)x
(x means times)
-1 x +2 x +2 x +2 x +2
4
+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
4
(-2) = - 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.
0
2 =1
0
4,638 =
1
0
Any number(except the number 0) = 1
0
0 = undefined
When the exponent is 1,
the answer is the same number as the base number.
1
2 =2
4,6381 = 4,638
any number1 = the same
base “any number”
1
0 =0
The exponent 1
is
usually
invisible.
The invisible exponent 1
1
2 =2
1
4,638 =
4,638
1
any number = the same
base “any number”
1
0 =0
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
5
=
means 2
2x2x2x2x2
product = write the long way = answer
“Find the value of the product…”
means
answer
5
2 =
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
3
125 = power 5 = 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