Transcript Thinking Mathematically by Robert Blitzer

P.2 Exponents and
Scientific Notation
Definition of a Natural Number
Exponent
• If b is a real number and n is a natural
number,
n
b  b  b b ...b
• bn is read “the nth power of b” or “ b to the
nth power.” Thus, the nth power of b is
defined as the product of n factors of b.
Furthermore, b1 = b
• “Special” powers are 2 (squared)
and 3 (cubed).
Find the exponent button on your calculator.
The Negative Exponent Rule
• If b is any real number other than 0 and n is
a natural number, then
b
n
1
 n
b
The negative exponent “flips” the base to the other side
of the division bar to become a positive exponent. It
DOES NOT CHANGE the SIGN of the base!
Ex: -2
3x-5
ans:
2x

3
5
The Zero Exponent Rule
• If b is any real number other than 0,
b0 = 1.
Ex: (for x not equal to zero)
(56892302 x
79
4378 0
)  1
The Product Rule
b m · b n = b m+n
When multiplying exponential expressions with the
same base, add the exponents. Use this sum as the
exponent of the common base.
Hint: if you get these rules confused, think of a
simple example and work it manually. Such as:
x  x  xx  xxx  xxxxx  x or x
2
3
5
(23)
The Power Rule (Powers to Powers)
(bm)n = bm•n
When an exponential expression is raised to a power,
multiply the exponents. Place the product of the
exponents on the base and remove the parentheses.
Hint: if you get these rules confused, think of a
simple example and work it manually. Such as:
( x )  ( x )  ( x )  ( x )  xx  xx  xx 
2 3
2
2
xxxxxx  x 6 or x (23)
2
The Quotient
Rule
m
b
m n
n b
b
• When dividing exponential expressions with the same
nonzero base, subtract the exponent in the denominator
from the exponent in the numerator. Use this difference as
the exponent of the common base. (Shortcut: subtract
“up” or “down” depending on which is the smaller
exponent.)
Hint: if you get these rules confused, think of a simple
example and work it manually. Such as:
x 5 xxxxx

 xxx (two x's divide out)
2
x
xx
=x 3 or x (52)
Q: Can you also think of an example that would demonstrate
the zero exponent rule using the quotient rule?
Ex: Find the quotient
3
a)
4
2
4
b)
14x7
10x10
Ans: a) 1

1
or
5
4
1024
b)
7
5x3
Products to Powers
(ab)n = anbn
When a product is raised to a power, raise
each factor to the power.
Hint: if you get these rules confused, think of
a simple example and work it manually.
Example
Simplify: (-2y)4.
Solution
Long way: (-2y)4
=(-2y)(-2y)(-2y)(-2y)
=(-2)(-2)(-2)(-2)(y)(y)(y)(y)
(by commutative law)
= (-2)4y4
= 16y4
Short way: (-2y)4
= (-2)4y4
= 16y4
(by “products of powers”)
Now you try to simplify each of the following, then
check below. (Hint: one of these cannot use any of
the shortcut rules we have discussed, why?):
-(-3x2y5)4
(x+2) 2
Ans:
81x8 y 20
and
x 2  4 x  4 NOT x 2  4
The second one has ADDITION, our rules refer to mult or divisn.
Quotients to Powers
n
a
a
   n
b
b
n
• When a quotient is raised to a power, raise
the numerator to that power and divide by
the denominator to that power.
Example
• Simplify by raising the quotient (15x7/6)-4 to
the given power.
Solution:
 15 x 
 6 


7
4
Ans: (Hint: reduce inside parenthesis first!)
16
625x 28
Properties of Exponents
1. b
n
1
 n
b
bm
5. n  b m n
b
0
m
n
mn
2. b  1 3. b  b  b
6. (ab)n  a n b n
m n
4. (b )  b
a n a n
7.    n
b  b
mn
Do problem #62 p 22.
 30a b 
 10a17 b 2 


14 8
3
3
30
 3b 
27b
 9
Ans: (Again, inside parenthesis first)  

3
a
 a 
10
(optional) Scientific Notation
The number 5.5 x 1012 is written in a form called scientific notation. A
number in scientific notation is expressed as a number greater than or equal
to 1 and less than 10 multiplied by some power of 10. It is customary to use
the multiplication symbol, x, rather than a dot in scientific notation.
Example
• Write each number in decimal notation:
a. 2.6 X 107
b. 1.016 X 10-8
Solution:
a. 2.6 x 107 can be expressed in decimal notation by moving the
decimal point in 2.6 seven places to the _______.
2.6 x 107 = ________________________.
b. 1.016 x 10-8 can be expressed in decimal notation by moving the
decimal point in 1.016 eight places to the _______.
1.016 x 10-8 = _______________________.
Scientific Notation
To convert from decimal notation to scientific notation, we reverse the procedure.
• Move the decimal point in the given number to obtain a number greater than
or equal to 1 and less than 10.
• The number of places the decimal point moves gives the exponent on 10; the
exponent is positive if the given number is greater than 10 and negative if the
given number is between 0 and 1.
Example:
Write each number in scientific notation.
a.
4,600,000
ans:
b.
0.00023
ans:
4.6 x 106
2.3 x 10
-4