10.1 – exponents notation that represents repeated multiplication of the same factor. where a is...
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10.1 – ExponentsNotation that represents repeated multiplication of the
same factor.
where a is the base (or factor) and n is the exponent.
Examples:
;na
34 4 4 4 512 12 12 12 12 12 7r r r r r r r r
24 4 4 16 24 4 4 16
24 4 4 16
Product Rule for Exponents
If m and n are positive integers and a is a real number, then
Examples:3 54 4
m n m na a a
4 4 4 4 4 4 4 4 3 54 84
3 27 7 7 7 7 7 7 3 27 57
8 69 9 149 3 7 93 3 3 193
6 2 3s s s 11s 7m m 8m
10.1 – Exponents
Power Rule for Exponents
If m and n are positive integers and a is a real number, then
Examples:
423
nm m na a
2 2 2 23 3 3 3
2 43
83
1049 4 109 409
36z 6 3z 8 2y
83
18z 28y 16y
10.1 – Exponents
Power of a Product Rule
If m, n, and r are positive integers and a and b are real numbers, then
Examples:
424y
rm n m r n ra b a b
33x
1 4 2 44 y
327x
34 22 p q r
4 84 y
3 33 x
1 3 4 3 2 3 1 32 p q r
3 12 6 32 p q r 12 6 38p q r
8256y
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
Examples:
26
3
5
9
x
y
rm m r
n n r
a a
c c
442
3
x
y
1 4 4 4
1 4 1 4
2
3
x
y
4 16
4 4
2
3
x
y
16
4
16
81
x
y
1 2 6 2
1 2 3 2
5
9
x
y
2 12
2 6
5
9
x
y
12
6
25
81
x
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
Examples:
mm n
n
aa
a
5
3
x x x x x x
x x x x
x x x
x x x
x x
2x x x
5
3
x
x 5 3x 2x
10.1 – Exponents
Quotient Rule for Exponents
Examples:9
6
5
57 3y 4y
7
3
y
y
9 65 35
14 102
42
14
10
2
2
4 1 11 17a b 3 107a b4 117a b
ab
16
10.1 – Exponents
What is the Rule?8
8
y
y
4
4
6
6
y
y
k
k
9
9
5
5
x
x
8 8y 0y 1 4 46 06 1
y yk 0k 1
9 95x
05x 1
10.1 – Exponents
Zero Exponent0 1, as long as 0.a a
If a is a real number other than 0 and n is an integer, then
Problem:
1nn
aa
3
5
x x x x
x x x x x x
x x
x x x
x x x
2
1 1
x x x
3
5
x
x 3 5x 2x
2x2
1
x
10.2 – Negative Exponents
Examples:
35 3
1
58x 8
1
x
47k 4
7
k 4
3
4
1
3
1
81
1 15 3 1 1
5 3
3 1 5 1
3 5 5 3
3 5
15 15
8
15
10.2 – Negative Exponents
If a is a real number other than 0 and n is an integer, then
Examples:
1 1n nn n
a and aa a
4
1
x 0
4
x
x 0 4x 4x
6
x
x 1 6x 7x
10.2 – Negative Exponents
Examples:5
6
y
y
5 6y 11y
9
2
r
z
2
9
z
r
26
7
2
2
6
7
2
2
7
6
49
36
10.2 – Negative Exponents1 1n n
n na and a
a a
Practice Problems
35
4
x x
x
10
45
y
y
239x
y
15
4
x x
x
16
4
x
x 16 4x 12x
10
45
y
y
10
20
y
y
10
20
y
y
10 20y 30y
410 5
1
y y
10 20
1
y y
30
1
y
2 6
2
9 x
y
2
2 69
y
x
2
681
y
x
10.2 – Negative Exponents
Practice Problems
54 7a b
3 6
5 2
32
8
x y
x y
20 35a b 20
35
a
b
3 5 6 24x y 8 44x y 8
4
4x
y
3 6
5 2
32
8
x y
x y
3 5 2
6
4x x y
y
8
4
4x
y
4 5 7 5a b
10.2 – Negative Exponents
Scientific Notation
A number is written in scientific notation if it is a product of a number a, where and an integer power r of 10.
Examples:
367,000,000
10 10a
165 83.67 10
0.00017
21.65 10
41.7 10
0.00597 35.97 10
10ra
10.2 – Negative Exponents
Scientific Notation
Examples:
0.000009621
27,500
69.621 10
5,420
42.75 10
35.42 10
0.0000000000735117.35 10
10.2 – Negative Exponents
Definitions
Coefficient: the numerical factor of each term.
Constant: the term without a variable.
Term: a number or a product of a number and variables raised to a power.
Polynomial: a finite sum of terms of the form axn, where a is areal number and n is a whole number.
2 23, 5 , 2 , 9x x x y
2 29, ,5 2x x x y
3, 6, 5, 32
3 215 2 5x x 6 5 321 7 2 6y y y y
10.3 – Polynomials
Definitions
Monomial: a polynomial with exactly one term.
Binomial: a polynomial with exactly two terms.
Trinomial: a polynomial with exactly three terms.
8,x
2 ,ax
2 8,x x
9 ,m 29x y42 ,x,rt
3,r 25 2 ,x x 22 9x x y
5 3 3,r r 25 2 7x x
10.3 – Polynomials
Definitions
25x
The Degree of a Term with one variable is the exponent on the variable.
The Degree of a Term with more than one variable is the sum of the exponents on the variables.
27x y The Degree of a Polynomial is the greatest degree of the terms
of the polynomial variables.32 3 7x x
2, 42x 14, 9m
3, 4 22x y 106, 5 49mn z
4 2 2 32 5 6x y x y x 63,
10.3 – Polynomials
Practice Problems
3 25 4 5x x x Identify the degrees of each term and the degree of the polynomial.
2 4 5 33 2 9 4a b ab b
5 4 4 5 3 34 5 6 2x y x y x y xy
3 2 1
3
9 9 6 2
9
6 6 3 0
6
10.3 – Polynomials
Combining Like Terms - Practice Problems
2 214 3 10 94y y Simplify each polynomial.
223 6 15x x x
24y 91
223x 7x 15
10.3 – Polynomials
Practice Problems
Simplify each polynomial.
3 32 1 1 32
7 4 2 8x x x x
3 32 1 1 32
7 2 4 8x x x x
3 32 1 1 32
7 2
2 7 2
2 7 4 82x x x x
3 34 7 2 32
14 14 8 8x x x x
33 12
14 8x x
10.3 – Polynomials
Practice Problems
23 10 1y for y
Evaluate each polynomial for the given value.
26 11 20 3x x for x
23 11 0 3 1 10 3 10 7
26 11 23 3 0
6 9 33 20
54 33 20 87 20 67
10.3 – Polynomials