Download Section 7.4 Integer Exponents and Scientific Notation

Section 7.4 Integer Exponents and Scientific Notation
Exponent - A number or symbol, as 3 in (x + y)3, placed to the right of and above another number, variable, or expression
(called the base), denoting the power to which the base is to be raised. Also called power.
The exponent (or power) tells how many times the base is to be multiplied by itself.
Example 1:
(x + y) = (x + y)(x+y)(x+y)
3
Example 2:
(-3) = (-3)(-3)(-3)(-3) = 81
4
Properties of Exponents
If m and n are integers, then
If n is a positive integer (-n is negative) , and x ≠ 0 , then
xm xn = xm+n
x −n =
Example: (x2)(x4) = x6
1
xn
1
= xn
−n
x
Examples :
If m and n are integers, then
( x m ) n = x m⋅n
5 x −2 =
Example: (x5)2 = x10
If x is a real number and
0
and
5
x2
3
= 3x 2
−2
x
x ≠ 0 , then
x =1
n
an
 a
= n
If n is an integer and b ≠ 0 , then  
 b
b
0
Example: 5 = 1
If m, n, and p are integers, then
Example :
( xy ) n = x n y n ,
2
x2 x2
x
  = 2 =
3
9
3
Example: (xy)4 = x4y4
and
m
n
p
(x y ) = x
m⋅ p
y
n⋅ p
Example: (x3y2)4 = x12y8
If m and n are integers and x ≠ 0 , then
xm
= x m− n
n
x
x5
5− 2
3
Example : 2 = x = x
x
If n is a positive integer (-n is negative) , and b ≠ 0 , then
−n
n
 a
 
 b
 b
= 
 a
Example :
x
 
3
−2
32
9
= 2 = 2
x
x
Section 7.4 Integer Exponents and Scientific Notation
Example 3 p. 394 Simplify
A.
B.
− 35a 6b −2
25a −2b 5
(− 2 x )(3x −2 )−3
SCIENTIFIC NOTATION
Scientific Notation is a way to describe very large or very small numbers.
For example there are an estimated 1,000,000,000,000,000,000,000 stars in the
universe.
That’s a billion trillion stars! An easier way to write that number is to use scientific
notation.
Start from the decimal point (or the end of the number if there is no decimal point),
and count how many spaces you go to the left until you get to just to the right of the
first digit.
1,000,000,000,000,000,000,000 = 1. X 1021 = 1021
Scientific Notation (exponential notation) refers to writing decimal numbers as a number
multiplied by a power of 10. Now this is done in a certain way. What this boils down to is
moving a decimal point, keeping track of how many places you moved it, and increasing or
decreasing the power of 10 by that number of decimal point moves.
Formally now, an exponential number has two parts, a mantissa and an exponent. The
mantissa is a number whose magnitude is greater than or equal to 1 but less than 10. The
exponent is a power of 10.
Here's an example: 2.43 x 103
the mantissa is 2.43 the exponent is 3
Section 7.4 Integer Exponents and Scientific Notation
Here's another example: -5.104 x 10-2
the mantissa is -5.105 and the exponent is -2
Ok, so what are these numbers?
2.43 x 103 = 2430 and -5.104 x 10-2 = -0.05104
Here's the deal. The exponent is the number of decimal places to move left or right to
remove the power of 10. How's that? Ok, let's consider a few examples.
Example:
Take the number 132.45, we want to write this number in scientific notation.
First of all this number is greater than 9.999.... so we need to rewrite it using powers of ten.
We need to move the decimal point. Let's move it right by 1. We get 1324.5, and we're
getting a number even larger than 9.999.... So let's move it left by 1. We get 13.245, we're
getting closer to 9.999... but we no longer have the same number. 13.245 ≠ 132.45, but if we
multiply 13.245 by 10 we'll have the same number. We have 13.245 x 101.
132.45 = 13.245 x 10 (we usually don't show the 1 in the exponent)
Well, 13.245 is still greater than 9.999... so, let's move the decimal left one more time.
We get: 1.3245 for the mantissa and this number is in the correct range but we now need to
multiply by 100 = 102 to keep the same number.
So we have 132.45 = 1.3245 x 102
(notice, we moved the decimal left twice and increased the exponent of ten by 2)
Example:
consider 0.000716
The mantissa must be greater than or equal to 1 and less than 10, and this number is
much to small. If we move the decimal right by 1 we get 0.00716 but we need to multiply
by 10-1 to keep the same number. So if we move the decimal right by 4 we get 7.16 which is
in range for the mantissa but we need to multiply by 10-4 to keep the same number. So what
we have is
0.000716 = 7.16 x 10-4
(we moved the decimal right 4 times and decreased the exponent of ten by 4)
The point of scientific notation is the ability to write very large or very small numbers in a
compact form.
Examples of very large numbers are:
Avogadro's Number: 6.02257 x 1023 (number of atoms of a substance to give the number of
grams equal to the atomic mass of the substance)
speed of light: 3.0 x 108 meters per second
Section 7.4 Integer Exponents and Scientific Notation
googool = 1x10100
diameter of an atom = 2.5 x 10-10 meters
Operations with numbers in exponential notation. A review of exponents would be very
helpful here before proceeding with the examples.
Multiplication/Division
This is easier, so we'll do this first.
Procedure:
Multiply/divide the mantissas, then the powers of 10, then write the result in proper form.
Examples:
1) 8.3x104 X 2.0x105 = (8.3 x 2.0) X ( 104 x 105 ) = 16.6 x 109
now, rewriting, we get 1.66x1010
(1.2× 10 ) =  1.2  ×  10
2)
(4.0 × 10 )  4.0   10
−1
−1
5
5

 = 0.3 × 10 −1=5 = 0.3 × 10 −6

That mantissa is not greater than or equal 1 so we multiply the mantissa by 10 and divide
and subtract 1 from the exponent. We then get 3.0 x 10-7
Addition/Subtraction
The exponents of the addends or subtrahends must be the same.
Procedure: Adjust the exponents until they are the same; as you do this you will be
adjusting the mantissas as well. Combine mantissas and rewrite the result in proper
exponential form. By doing this you are creating “like terms.” Jus pretend that the 10’s are
x’s.
Example: 3.21x103 + 8.41x102
we'll do this two ways. First, we'll change the first number to an exponent of 2
32.1x102 + 8.41x102 = 40.52x102 = 4.052x103
now let's write the second number to have an exponent of 3
3.21x103 + 0.841x103 = 4.052x103
either method is fine.