10.1 – Exponents Notation that represents repeated multiplication of the same factor. n a ; where a is the base (or factor) and n.
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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:
4 444
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 mn
Examples:
43 45 4 4 4 4 4 4 4 4 435 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
mn
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
63
4 10
8 2
9
y
410
82
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
mr
a b
m n
n r
Examples:
3
27x
3x 3 x
3
3
4y
2 4
3
4 y
14
2 p q r
4
3
2
2
3
24
4 y 256 y
4
2
p qr
12
8
6 3
13
43
8
23 13
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 mr
n nr
c
c
m
Examples:
4
2x
214 x 44
14 14
3 y
3y
4
2
5x
512 x 62
3 12 32
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
mn
a
n
a
x5 x x x x x
xxxxx
2
x
x
x
3
x
xxx
xxx
x5
53
2
x
x
3
x
10.1 – Exponents
Quotient Rule for Exponents
Examples:
59
3
96
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 1 111
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
44
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
xxx
xxx
2
x x x
x5 x x x x x
xxxxx
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
31 51
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
16
7
x
x
6
x
10.2 – Negative Exponents
a
n
1
n
a
and
Examples:
y 5
56
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
92 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
35 62
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
9.621 106 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 x2 x x
7
4
2
8
2 3 1 3 1
3
x x x x2
7
2
4
8
2 2 3 7 1 3 2 1
3
x x x x2
2 7
7 2
2 4
8
4 3 7 3 2
3
3 3 1
x x x x2 x x2
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