Download 1 Guided Notes for lesson P.2 – Properties of Exponents If a, b, x, y

Guided Notes for lesson P.2 – Properties of Exponents
If a, b, x, y   and a, b,  0 , and m, n  Z then the following properties hold:
1. Negative Exponent Rule: b  n 
1
1
and  n  b n Answers must never contain negative exponents.
n
b
b
Examples:
a) 53
b)
1
4 2
2. Zero Exponent Rule: b0  1
Examples:
a) 7 0
b)
 5
c) 50
0
3. Product Rule: bm g bn  bm  n - keep the base and ________ the exponents
Examples:
a) 22 g 23
4. Quotient Rule:
b) x 3 g x 7
bm
 b m  n - keep the base and ____________________ the exponents.
n
b
Examples:
212
a) 4
2
b)
1
x2
x5
5. Power to a Power Rule  bm   bm n - keep the base and __________________ the exponents.
n
Examples:
a)
2 
2
5
b)
x 
3 4
6. Product to a power:  ab   a nbn - raise both bases to the same power.
n
Note well:  a  b   a n  bn and  a  b   a n  bn
n
n
Examples:
a)
 2 y 
b)  3x 2 y 5 
4
n
2
an
a
7. Quotient to a power:    n - raise both bases to the same power.
b
b
Examples:
 2 
a)  
 5y 
4
 x 3 
b)  2 
 3y 
2
2
Simplifying Exponential Expressions
1) No parentheses.
2) No powers raised to a powers.
3) Each based occurs only once.
4) No negative exponents.
5) Simplify numerical expressions.
Examples: Simplify the expression.
a)
 2x y 
b)
 3x y 
c)
 7 xy  2 x y 
d)
 6x y 3xy 
e)
35 x 2 y 4
5 x 6 y 8
f)
100 x 2 y 6
20 x 4 y 3
8
4 2
4
 4x 2 d 3 
g)  7 4 
y g 
5
6
3
4
2
 5 x 8 
h)  4 
 2y 
3
5 3
5
5
3
Definition: A number is written in _____________________________ if it is of the form _____________
where 1  a  10 and n  Z .
Examples : Write in standard form (decimal form).
a) 6.4  105
b) 1.25  107
c) 2.17  103
d) 6.018  105
Example 2: Write in scientific notation.
b) 0.000000000000802
a) 34,970,000,000,000
 2  10    4  10 
 2  4   10  10 
6
To multiply numbers written in scientific notation:
1. Multiply the decimal parts first. (use your calculator, if necessary)
2. Using the rules of exponents, multiply the powers of ten. (add the exponents)
3. Be sure your final answer is in scientific notation.
a)
3  10    2  10 
5
7
d) 8  103    4  105 
b)  5  103    3  105 
e)
9  10   3  10 
7
8
4
7
6
8  1013
c)
 6  10    7  10 
f)
 6  10    2.3  10 
2
6
8
4
7
 4  10    2  10 
 4  2   10  10 
15
To divide numbers written in scientific notation:
1. Divide the decimal parts first. (use your calculator, if necessary)
2. Using the rules of exponents, divide the powers of ten. (subtract the
exponents)
3. Be sure your final answer is in scientific notation.
15
2  108
Examples: Find the quotient, express answers in scientific notation.
a)
8  105
2  103
b)
6  1012
3  108
c)
9  104
3  107
d)
8.24  1024
5.15  1018
e)
3  105
6  108
f)
2  103
5  1012
5
7
7
Guided Notes for lesson P.3 – More with Exponents
If b   and b  0 , and m, n  Z then the following properties hold:
1
If n is even, b  0
8. Fractional Exponents: b n  n b 
 If n is odd, b 
1
1 If n is even, b  0
bn  n 
b  If n is odd, b 
and
Examples:
a) 64
1
2
b) 125
1
3
c) 16
1
d)
 27  3
e) 64
m
9. Fractional Exponents: b  b 
n
n
m
 b
n
m
 If n is even, b  0
and b n 

 If n is odd, b 
m
1
n
bm

1
4
1
3
1
 b
n
m
Examples:
a) 27
2
3
3
b) 4 2
c) 8
5
3
4
3
4
e)
d) 81
6
 1000  3
 If n is even, b  0

 If n is odd, b 
Examples: Simplify the expression. Express your answer using a radical when your final answer yields a
fractional exponent.
 1  3 
a)  5 x 2  7 x 4 



 5 1
b)  2x 3 y 6 


7
c)
 14 32
x y
d)  3 7

 x2 y4

32 x 3
1
16 x 4
e)
 x  x 
3
f)
7
6
12





 x  x 
4
5
3
7
Examples: Factor the expression and simplify. This is a skill in preparation for calculus.
3
a)
 x  1 2

1
3x
 x  1 2
2
1
4
b)  x 2  4  3 
1
c) 2 x 8  x 2  2  8  x 2  2
8
1
8x 2
x

4

3
3
Guided Notes for lesson P.7 – Absolute Value Equations
Absolute Value:
Definition: a means the distance the number a is from ________ on a number line.
Solving Absolute Value Equations:
If X is any algebraic expression and c  Z  , then the solutions to X  c are found by solving the equations
_______ and _______.
Example Solve the equation.
a) x  5
b) x  3  7
c) x  8  4
d) 2 x  3  15
e)
f) 4  5 x  7
1
x 7  4
3
9
Guided Notes for lesson P.9 – Inequalities and Absolute Value Equations and Inequalities
To solve an inequality means to find all the possible values of the variable that makes the inequality true.
To solve an inequality you use the same inverse operation(s) to both sides of the inequality with one slight
glitch. Keep in mind you want the variable all alone on one side of the inequality.
When an inequality is multiplied or divided by a positive number the inequality sign remains unchanged.
Example to illustrate:
50  35 true
5  6 true
50 35
2  5   2  6  multiply by 2

divide by 5
5
5
10 < 12 true
10 < 7 true
When an inequality is multiplied or divided by a negative number the inequality sign is reversed.
Example to illustrate:
5  6 true
50  35 true
 2  5    2  6  multiply by  2
50
35

divide by  5
5
5
10 <  12 false
10 >  7 false
10  12 true
10   7 true
Examples: Solve each of the following and graph your solution on a number line.
a)
x5  2
c)
4 n  16
b)
d)
10
3  x  5
n
 8
2
e)
x
 22
10
f)
13x  208
g)
6 x  15  57
h)
4 x  40   7 x  65
i)
12  2 x  13  117  15 x
j)
3  5x  2  9
i) 2  1  3x  8
11
Three ways to display solutions to inequalities:
Set builder Notation
(used for Alg 2)
Interval Notation
(used in pre-calc on up)
xaxb
xaxb
xxa
xxa
xxb
xxb
x x
12
Graph
(used for either Alg 2 or pre-calc on up)
Connectors of sets of numbers:
Set builder Notation Symbol
Interval Notation Symbol
Intersection (and)
(what is in both)
Union (or)
(what is in either or both)
Set builder Notation
Interval Notation
x 1  x  4  2  x 8
x 1  x  2  7  x 9
13
Graph
Solving Absolute Value Inequalities
Solution: If X is any algebraic expression and c  Z  , then the solutions to X  c are found by solving
_______ and _______.
Examples: Solve the inequality and graph the solution set.
a) x  5  4
b) 2 x  3  1
c) 2 5  3x  7  13
14
Solution: If X is any algebraic expression and c  Z  , then the solutions to X  c are found by solving
__________ and __________.
Examples: Solve the inequality and graph the solution set.
a) x  7  10
b)
1
x 3  8
4
c) 5  2 x  7
15
Guided Notes for lesson P.3A – Radicals
Definition:
n
Date: _________
a is called the _____________________ nth root or _____________________.
a is called the ________________ and n is called the ________________.
Note Well: If n is even, a  0. If n is odd, a .
If a, b   and b  0 , and m, n  Z then the following properties hold:
1. Product Rule:
n
2. Quotient Rule:
n
3. Power Rule 1:
n
4. Power Rule 2:
n
To simplify
n
ab  n a gn b .
a

b
n
n
Note Well
n
 a  b 
n
a  n b and
n
 a  b 
n
a nb
a
b

 _____________________________

an  
 _____________________________


a a
m
m
n
a m _______________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
16
Example: Simplify the radical. Express the answer in simplest radical form.
a)
32
b)
d)
3
48
e)
g)
3
250x 4
h)
j)
3
75
c)
375
f)
3
686
50x 5 y 6
i)
5
64x 9 y12
49 x 5
64 y 9
k)
700
3
54 x 6
8 y7
Definition: Like radicals are radicals with the same _____________________ and ___________________
Rule: You may only add or subtract like radicals.
17
Example: Simplify the expression.
a) 8 3 + 5 3
b) 7 3 5  113 5
c) 8 12  3
d)
3
8x 4  3  x  4 3 27 x
Example: Multiply. Express the answer in simplest radical form, if necessary.
a) 8 3 g
c)
1
15
2
 2  5 3  5 

b) 3 2 2 8  3
d)

 4  7  4  7 
Definition: The irrational numbers a  b and a  b are called _______________________ and when
multiplied together will always yield a _________________ number. Why?
18
e)
10  2 
2
2  3
3
f)
Example: Divide. Express the answer in simplest radical form, if necessary.
48 54
a)
12 3
c)
4 45  6 60
2 5
10 3 48
b)
53 2
d)
19
15  180
5
Definition: To _____________________________________________________ means to find an
equivalent fraction with a denominator that is a rational number. Why was/is that important?
1
2
2
2
Example: Rationalize the denominator.
a)
c)
e)
2
5
b)
2
4  11
d)
3  10
2  10
f)
20
10
3
3
3
3 1
x  h x
x  h x
Guided Notes for lesson 10.5 – The Binomial Theorem
Definition: A _______________ is a two termed algebraic expression.
Definition: When any binomial is raised to a positive integral power, the result is called an ____________
Illustration: Expand  x  y 
A few things you should notice in the expansion of  x  y  :
3
3
1) the x’s decrease in power  x 3 , x 2 , x1 , x 0  term by term.
 x  y 3   x  y  x  y  x  y 
  x  y   x 2  2 x1 y 1  y 2 
2) the y’s increase in power  y 0 , y1 , y 2 , y 3  term by term.
 x 3  2 x 2 y1  xy 2  x 2 y1  2 xy 2  y 3
 x 3  3 x 2 y 1  3 x1 y 2  y 3
3) the exponents on x and y always add up to 3 for each term.
4) the number of terms (4) is one greater than the exponent 3.
5) there are coefficients on the two middle terms
Where the coefficients of a binomial expansion come from?
Definition: The coefficient of any term of binomial expansion is called a binomial coefficient and is found
n
n!
by   
provided n, r ¢  and n  r .
 r  r ! n  r !
n
C is used as well to denote   . A combination of n things taken r at a time.
n r
r
Examples: Evaluate by hand
 3
a)  
2
8
b)  
 5
7
c)  
1
Example: Evaluate with your calculator
9
a)  
9
 12 
b)  
7
 5
c)  
 0
21
The Binomial Theorem: For any monomial expression a any monomial expression b, and n, r ¢  :
 a  b
n
 n  n 0  n  n 1 1  n  n 2 2
 n  2 n 2  n  1 n 1  n  0 n
    a   b      a   b      a   b   ...     a   b      a   b      a   b 
 0
1
2
n
n
n
Examples: Expand the expression. (write small)
a)  x  3
b)
4
 x  2
remember that  x  2    x  ( 2) 
3
3
3
c)  2x  y  make sure it’s the 2 and the x that are raised to the exponents.
5
22
d)
 3x  4 y 
e)
x
2
3
 6 y3 
4
Consider  x  y   x 3  3x 2 y  3xy 2  y 3
3
Another way to aid you with expansion of  x  y  is to use Pascal’s Triangle
1
1
1
3
1
1
2
3
1
3
1 x 3  3x 2 y  3xy 2  y 3
Example: Expand  x  3 y  using Pascal’s Triangle.
5
23
Consider the expansion of  x  y   x 3  3x 2 y  3xy 2  y 3 .
If we just wanted the second term of its expansion without expanding it, how could we find it?
 3  31
1
It would be    x   y   3x 2 y
1
3
For any monomial expression a any monomial expression b, and n, r ¢  The rth term of a Binomial
Expansion without expanding is:
 n  n  r 1 r 1
b
 r  1  a 


Examples: Find the given term of the epansion with out expanding it.
a)
 x  2
c)
 3x  5
e)
x
3
5
(third term)
6
(sixth term)
b)
 x  6
d)
 2 x  7
 2 y 5  (third)
4
24
8
(fourth term)
9
(second term)
Good luck to: _____________________________
GMP 4: Chapter PA Test: Sections: 10.5, P.2, P.3, and P.9
Multiple Choice Questions: For 1-18, circle the best answer to the question. When you see
the saw graphic, SHOW ALL WORK in the space provided.
 Correct answers with no work will receive minimal credit.
 Incorrect answers with work will receive partial credit.
 Incorrect answers with no work will receive no credit.
You may use a calculator. Each question is worth 9 points.
1) What is the solution of the inequality x  3  5 ?
a)
x 8  x  2
b)
x 2  x  8
c)
x x  8or x  2
d)
x x  2 or x  8
2) What is the solution of the inequality 2 x  3  4 ?
1
 7
a)  x  x   
2
 2
1

b)  x   x 
2

7

2
1

c)  x x  or x 
2

7

2
3) Which graph represents the solution to the inequality 1  2 x  7 ?
a)
b)
c)
d)
25
7
1

d)  x x  or x   
2
2

4) Which graph represents the solution to the inequality 2 x  3  7 ?
a)
b)
c)
d)
 10 
5) Evaluate the binomial coefficient:   .
5
a) 120
b) 252
c) 30,240
d) 3,628,800
c) 2
d) 1
4
6) Evaluate the binomial coefficient:   .
1
a) 24
b) 4
7) Expand  5x  2  using the binomial theorem.
3
a) 125x 3  8
b) 125x 3  150 x 2 16 x  8
c) 125x 3  150 x 2  60 x  8
d) 125x3  30 x 2  30 x  8
26
8) Expand  x 2  2 y  using the binomial theorem.
4
a) x8  8x6 y  24 x 4 y 4  8x 2 y 3  16 y 4
b) x8  2 x6 y  24 x 4 y 2  16 x 2 y 3  16 y 4
c) x 4  8x3 y  24 x 2 y 2  32 xy 3  16 y 4
d) x8  8x6 y  24 x 4 y 2  32 x 2 y 3  16 y 4
9) The fifth term in the expansion of  2x  y  is
6
a) 240x 2 y 4
c) 60x 2 y 4
b) 240x 2 y 4
d) 60x 2 y 4
9) The last term in the expansion of  x  3 y  is
4
a) 108xy 3
b) 81y 4
c) 3y 4
d) 243y 5
10) The middle term in the expansion of  x  y  is
4
a) 4x 3 y
b) 4xy 3
c) 6x 2 y 2
27
d) 6x 2 y 2
11) Simplify
a) 50 10
12) Simplify
a) 9 20
500
b) 10 5
c) 5 10
d) 10 50
c) 6 10
d) 3 40
c) 20  6 3 6
d) 16 3 4
c) 5a 4bc 2 3ac
d) 5a 7c3 3ac
30 g 12
b) 12 10
13) Simplify 5 3 16  2 3 54
a) 16 3
14) Simplify
a) 15a 4bc 2 5a 5c3
b) 16 3 2
75a 9b2c5
b) 3a 4bc 2 5ac
28


15) Simplify 5  2 3 2  4
a) 11 2  14
16) Simplify
a) 1
b) 11 2  8
a) 12  3
c) 14 2  20
d) 4 2  14
c) 2  3
d) 5  3
c) 4  2 3
d) 6  2 3
128  72
8
b) 2
17) Simplify

12
3 3
b) 2  3
142 15
, where N is the
 7r
number of rotations per minute required to simulate earth's gravity, and r is the radius of the space station. If
the number of rotations per minute is 10, which of the following is a reasonable estimate for the radius of
the space station?
18) A Space station rotates in order to simulate gravity, such that N 
a) 135m
b) 5.4m
c) 10m
29
d) 44m
Free Response Questions: For 19-43. When you see the saw graphic, SHOW ALL WORK
in the space provided.
 Correct answers with no work will receive minimal credit.
 Incorrect answers with work will receive partial credit.
 Incorrect answers with no work will receive no credit.
You may use a calculator. Each question is worth 7.5 points.
For 19-22, simplify the expression completely with only positive exponents.
19)
 2x y  6x y 
6
2
30 x3 y11 z 6
20)
5 x5 y 8 z 5
5
20) _____________________
19) _____________________
21)
 4x
2
y8 z 9 
4
 12 x5 y 6 
22)  11 2 
 6x y 
21) _____________________
3
22) _____________________
23) Write 4.56  106 in standard form without the use of exponents.
23) ___________________
24) Write 6,400,000 in scientific notation.
24) ___________________
30
 6.0  10  2.0  10 
Evaluate
 4.0  10 8.0  10 
3
25)
5
7
2
(express answer in scientific notation)
25) _____________________
For 26-28, simplify and express answer in radical form, if necessary.
1

 1 
26)  10 x 3   5 x 2 



1
27)


1
8x 5
26) ___________________
28) 81x 4 y 10
72 x 4
1
2
28) ___________________
31
27) ___________________
For 29-34, solve the inequality. Graph your solution on a number line.
29)
n  24  9
31)
2.6 y  1.3
33)
4 p  1   9
30)
x  14  54
32)
8 
34)
2d  10  6  7  4d 
32
k
4
For 35-36, solve algebraically for x.
35)
x 6  8
35) _______________________
36)
2 5 x  1  10
36) _______________________
For 37-38, solve algebraically for x. Express your answer in interval notation.
37) x  7  1  4
38) 3  7 x  9
37) __________________________
38) __________________________
33
For 39-40, expand the expression completely.
39)
 3x  5 y 
4
39) ________________________________________________
40)
x
3
 y2 
3
40) ________________________________________________
34
41) Express in simplest radical form.
a)
48x5 y9
b)
a) ________________________________
3
40x 7 y8
b) _____________________________
42) Simplify the expression.
6 


a) 2 45  3 125  405
b)
a) _______________________
b) _______________________
c)
2  5 
2
c) _______________________
d)
11 9  11
3 5
2 5
d) _______________________
35
Bonus) Find the exact value of: 13 
2
7
3 2
(+5)
Bonus) ___________________
36