Transcript 10.1 – Exponents Notation that represents repeated multiplication of the same factor. n a ; where a is the base (or factor) and n.

10.1 – Exponents
Notation that represents repeated multiplication of the
same factor.
n
a ;
where a is the base (or factor) and n is the exponent.
Examples:
4  444
3
12  12 12 12 12 12
5
r  r r r r r r r
7
42  4  4  16
 4 
2
42    4  4  16
  4    4   16
10.1 – Exponents
Product Rule for Exponents
If m and n are positive integers and a is a real number, then
a m  a n  a mn
Examples:
43  45   4  4  4   4  4  4  4  4   435  48
7  7   7  7  7  7  7  7
3
2
98  96  914
s s s  s
6
2
3
3 2
5
7

33  37  39  319
11
m m  m
7
8
10.1 – Exponents
Power Rule for Exponents
If m and n are positive integers and a is a real number, then
a 
m n
a
mn
Examples:
8
3
3 3 3 3 
2
2
3 
2 4
z 
6 3
2
 3
z
2
8
 3
9 
18
 z
y 
2 4
63
4 10
8 2
 9
 y
410
82
40
9


y
16
10.1 – Exponents
Power of a Product Rule
If m, n, and r are positive integers and a and b are real
numbers, then
r
a b 
mr
a b
m n
n r
Examples:
3
27x
 3x   3 x 
3
3
 4y 
2 4
3
4 y
14
 2 p q r 
4
3
2
 2 
3
24

4 y  256 y
4
 2 
p qr 
12
8
6 3
13
43
8
23 13
p q r
8 p q r
12
6 3

10.1 – Exponents
Power of a Quotient Rule
If m, n, and r are positive integers and a and c are real
numbers (c does not equal zero), then
r
 a  a mr
 n   nr
c
c 
m
Examples:
4
 2x 
214 x 44

  14 14 
3 y
 3y 
4
2
 5x 
512 x 62
 3   12 32 
9 y
 9y 
6
24 x16

4 4
3 y
52 x12

2 6
9 y
16 x16
81y 4
25 x12
6
81 y
10.1 – Exponents
Quotient Rule for Exponents
If m and n are positive integers and a is a real number and a
cannot equal 0, then
m
Examples:
a
mn

a
n
a
x5 x  x  x  x  x
xxxxx
2
x

x

x



3
x
xxx
xxx
x5
53
2
x

x

3
x
10.1 – Exponents
Quotient Rule for Exponents
Examples:
59
3
96
5
5


6
5
7
7 3
4
y
y

y

3
y
 2 
10
 2 
14
 2 
14 10
  2   16
4
7a 4b11
4 1 111
3 10
 7a b  7a b
ab
10.1 – Exponents
What is the Rule?
y8
8 8
0

y

y
1
8
y
64
0
44
6
1
6

4
6
ky
y y
0

k

k
1
y
k
5x  
9 9
 5x 
9
5x 
9
  5x   1
0
Zero Exponent
a  1, as long as a  0.
0
10.2 – Negative Exponents
Problem:
1
1
x3
xxx
xxx
 2



x x x
x5 x  x  x  x  x
xxxxx
x3
3 5
2
x
x


5
x
x
2
1
 2
x
If a is a real number other than 0 and n is an integer, then
a
n
1
 n
a
10.2 – Negative Exponents
Examples:
1
5  3
5
3
7k
4
7
 4
k
x
8
 3
1
 8
x
4

1
 3 
4
1

81
31 51
1 1
3 5
 
5 3        
5 3
 3  5  5  3 15 15
8
15
1
1
10.2 – Negative Exponents
If a is a real number other than 0 and n is an integer,
then
1
1
n
n
a  n and

a
n
a
a
Examples:
1

4
x
x0
4
0 4
x
 x

4
x
x
16
7

x
 x
6
x
10.2 – Negative Exponents
a
n
1
 n
a
and
Examples:
y 5
56
11
y

y

6
y
9
r
2
z
z2
 9
r
2
2
6
6
   2 
7
7
49
7

2
36
6
2
1
n

a
n
a
10.2 – Negative Exponents
Practice Problems
x 
5 3
x
4
x
10
y
y 
5 4
y 10
y 
5 4
x15 x

4
x

10
10
y
y
10  20
30
 y

 y
20
20
y
y

1

y
2
x16
12
16  4
x
 x

4
x
10
y 
5 4
1
1
 10 20  30
y y
y
y2
y2
 9x 
92 x 6
 2 6

 
6
2
9 x
81x
y
 y 
3
10.2 – Negative Exponents
Practice Problems
a

4 7 5
b
 a
4 5 7 5
b
20 35
 a b
20
a
 35
b
8
6
4x
32 x y
35 62
8 4
4x
y

4x
y


4
5 2
y
8x y
3
3
6
3 5
2
8
32 x y
4x x y
4x

 4
5 2
6
8x y
y
y
10.2 – Negative Exponents
Scientific Notation
A number is written in scientific notation if it is a product of a number a,
where 10  a  10 and an integer power r of 10.
a 10r
Examples:
2
1.65

10
165 
8
3.67

10
367, 000, 000 
4
1.7

10
0.00017 
3
5.97

10
0.00597 
10.2 – Negative Exponents
Scientific Notation
Examples:
2.75 10  27,500
4
9.621 106  0.000009621
5.42 103  5, 420
7.35 10
11
 0.0000000000735
10.3 – Polynomials
Definitions
Term: a number or a product of a number and variables raised
to a power.
3, 5x 2 ,  2 x, 9 x 2 y
Coefficient: the numerical factor of each term.
5x 2 ,  2 x, 9 x 2 y
Constant: the term without a variable.
3,  6, 5, 32
Polynomial: a finite sum of terms of the form axn, where a is a
real number and n is a whole number.
15 x3  2 x 2  5
21y 6  7 y 5  2 y 3  6 y
10.3 – Polynomials
Definitions
Monomial: a polynomial with exactly one term.
2
ax ,
rt ,
4
2x ,
9m,
2
9x y
Binomial: a polynomial with exactly two terms.
x  8, r  3, 5 x 2  2 x, 2 x  9 x 2 y
Trinomial: a polynomial with exactly three terms.
x 2  x  8,
r 5  3r  3, 5 x 2  2 x  7
10.3 – Polynomials
Definitions
The Degree of a Term with one variable is the exponent on
the variable.
5x 2  2,
2x 4  4,
9m  1
The Degree of a Term with more than one variable is the
sum of the exponents on the variables.
7x y  3,
2
2x y  6,
4
2
9mn z  10
5 4
The Degree of a Polynomial is the greatest degree of the terms
of the polynomial variables.
2 x  3 x  7  3,
3
2 x y  5x y  6 x  6
4
2
2
3
10.3 – Polynomials
Practice Problems
Identify the degrees of each term and the degree of the polynomial.
3a b  2ab  9b  4
3 0
6
6
6
5x  4 x  5x
2
1
3
3
2 4
2
3
4 x5 y 4  5x 4 y5  6 x3 y 3  2 xy
9
9
6
9
2
5
3
10.3 – Polynomials
Combining Like Terms - Practice Problems
Simplify each polynomial.
14 y  3  10 y  94 
2
2
4 y 2 91
23x 2  6 x  x  15  23x 2 7x 15
10.3 – Polynomials
Practice Problems
Simplify each polynomial.
2 3 1
1 3 3
x  x2 x  x 
7
4
2
8
2 3 1 3 1
3
x  x  x x2 
7
2
4
8
2 2 3 7 1 3 2 1
3
 x   x   x x2 
2 7
7 2
2 4
8
4 3 7 3 2
3
3 3 1
x  x  x x2   x  x2
14
14
8
8
14
8
10.3 – Polynomials
Practice Problems
Evaluate each polynomial for the given value.
3 y 2  10
for y  1
3  1  10
2

6 x 2  11x  20
3 1 10  3 10  7
for x  3
6  3  11 3  20  6  9  33  20 
2
54  33  20  87  20  67