Transcript Exponents & Powers

WARM-UP EXERCISES
Insert grouping symbols to make the
sentences true.
1.) 7- 4 x 6 + 4 = 30
2.) 6 + 2 x 3 - 4 = 20
3.) 9 - 4 x 2 + 5 = 15
4.) 5 x 8 - 6  2 = 5
Exponents & Powers
• Exponents are used to represent repeated
multiplication.
• 46
4 is the base. The base is the number that is
being repeated.
6 is the exponent. The exponent tells you how
many times to repeat the base.
Exponents & Powers
• 46 - Read Four to the sixth power.
• 72 - Read Seven to the second power or
Seven squared
• 53 - Read Five to third power or
Five cubed
Exponents & Powers
• Use an exponent to represent the
expression:
88888
85
3x  3x  3x  3x
(3x)4
Exponents & Powers
• Evaluate:
34
3333
81
55
55555
3125
ORDER OF OPERATIONS
1.) Do operations inside grouping symbols.
2.) Evaluate exponents.
3.) Do multiplication or division in order
from left to right.
4.) Do addition or subtraction in order from
left to right.
ORDER OF OPERATIONS
3  42 + 1
3  16 + 1
48 + 1
49
23 - [ ( 45  15)3  9 ]2
23 - [ 33  9 ]2
23 - [ 27  9 ]2
23 - 32
23 - 9
14
Write in exponential form.
1.) p squared
2.) t  t
3.) 5 t  t t  t
Evaluate each power.
4.) 64
5.) 25
6.) 93
Evaluate using the order of operations.
7.) 6  3 + 2  7
8.) 10  (3 + 2) + 9
9.) [(7 – 4)2 + 3] + 15
VARIABLES IN ALGEBRA
Variable - a letter that is used to represent one
or more numbers.
Evaluating an Algebraic Expression:
1.) Write the algebraic expression.
2.) Substitute values for the variables.
3.) Simplify the numerical expression using
the order of operations.
Examples
8x + 9y when x = 8 & y = 12
8  8 + 9  12
64 + 108
172
5x + (3x + 8) when x = 4
5  4 + (3  4 + 8)
5  4 + (12 + 8)
5  4 + 20
20 + 20 = 40
Assignment
1.) (x + y)2
when x = 5 & y = 3
2.) m – n2
when m = 25 & n = 4
3.) (a – b)4
when a = 4 & b = 2
4.) (d – 3)2
when d = 16
5.) c3 + d
when c = 4 & d = 16
6.) 16 + x3
when x = 2
7.) 3 + 2x3
when x = 4
8.) 3r2 – 17
when r = 6
9.) x/7 + 16
when x = 21
10.) 6 + 2p2
when p = 5
Homework Answers
 P. 12
 P. 19
41.) 64
13.) 19
14.) 32
42.) 9
15.) 300
16.) 213
43.) 16
17.) 34
18.) 91
44.) 80
19.) 18
20.) 24
45.) 100
46.) 24
Real Numbers
 Natural Numbers– Positive whole
numbers.
0, 1, 2, 3, …
 Some people do not include 0 with the
set of Natural Numbers.
 Integers – Positive and negative whole
numbers.
… -3, -2, -1, 0, 1, 2, 3 …
 Rational Numbers – A number that can be
written as the quotient of two integers. A
rational number either terminates or is a
repeating decimal.
-2, ½, ¾, √4 and 1/3
 Irrational Numbers – A number that
cannot be written as the quotient of
two integers. Irrational numbers
cannot be written as repeating or
terminating decimals.
√5, 1/7, π
Examples
 Identify as a natural, integer, rational or
irrational number. You may have more than
one answer.
1.-22
2.5/7
3.√49
4.12
Adding Real Numbers
If the signs are alike, then add
the numbers together and keep
the sign of the largest number.
- 5 + (-4)
5+4=9
-9
If the signs are different, then
subtract the numbers and keep
the sign of the largest number.
-12 + 36
36 - 12
24
24
Examples:
 -7 + (-6)
-13
 2.9 + ( -3.6) + (-3)
2.9 + (-6.6)
6.6 - 2.9 = 3.7
-3.7
Use your addition rules to answer the
following problems.
1.) –25 + 57
2.) –45 + (-33)
3.) 66 + (-103)
4.) –12.5 + 32.4
5.) –23.6 + (-54.7)
6.) 54 + (-23) + 45
7.) –43 + (-67) + (-89)
8.) 2.6 + (-3.4) + 5.8
Subtracting Real Numbers
 To subtract real numbers, change the problem to
an addition problem and solve.
 SUBTRACTION TO ADDITION:
(Copy/Change/Change)
1.) Keep the first number the same.
2.) Change the subtraction sign to an addition sign.
3.) Change the number behind the subtraction sign
to its opposite.
4.) Follow the addition rules.
Examples:
-4-7
8 - (-6) - 7 + 14
- 4 + (-7)
8 + 6 + (-7) + 14
-11
14 + (-7) + 14
13 - (-35)
7 + 14
13 + 35
21
48
Use your subtraction rules to answer the
following problems.
1.) 75 – (-25)
2.) –54 – (-23)
3.) 76 – (-130)
4.) –52.1 – 43.2
5.) –6.32 – (-74.5)
6.) 45 – (-32) – 54
7.) –34 – (-76) + (-98)
Multiplying and Dividing
with positives and negatives:
1.) Multiply or divide the numbers first.
2.) Place your sign on your answer.
a.) If the signs are the same, then my
answer is positive.
b.) If the signs are different, then my
answer is negative.
DIVISION OF REAL NUMBERS
1. Divide the numbers. (If fractions
are involved, change the division
to multiplication and flip the
fraction behind the division sign.)
2. Place a sign on your answer.
(Same rules as multiplication.)
Examples
1. 51  (-3)
2. -216 ÷ (-12)
3. -5  2/3
Examples
4.
48
-½
3. -3x ÷ 2/5
Combining Like Terms
 Terms are separated by “+” and “-”
signs.
 Terms having the same variable and
same exponent can be combined.
Use Addition rules.
 Terms that do not have a variable
may be combined.
1.) 7x + 12x
19x
2.) 8y – 5y + 8
3y + 8
3.) -5x + x – 10 + 8
-5x + 1x – 10 + 8
-4x – 2
Homework Answers
8.) –12t + 30
9.) –3x + 6
10.) 11b + 12
11.) 3h – 3
12.) –10k + 10
13.) 15a – 5
14.) 9c – 5
15.) 1d – 3
16.) 3h – 7
17.) 8x + 7
18.) 4z + 5
19.) 2y + 12
20.) 8p
21.) 10m + 3
Adding and Subtracting
Polynomials
 STANDARD FORM OF A POLYNOMIAL:
 5x3 – 4x2 + 2x – 3
 *Always written in descending order, beginning
with the term with the highest exponent first
and ending with the constant or the term with
the smallest exponent.
 Coefficient – The numbers that come before
the variables.
Steps for Adding & Subtracting
1. Get rid of parentheses. (put a “1”
in front of parentheses that do not
have a number and multiply each
term.)
2. Combine your like terms.
3. Write in standard form.
Warm – Up Exercises
1.) 3 + 8  3 ÷ 2
2.) 3 • 23 + 4  6
3.) 5  (12 + 3  4)
Properties of Addition & Multiplication
• Associative Property of Addition
(a + b) + c = a + (b + c)
• Associative Property of Multiplication
(a • b) • c = a • (b • c)
• In the associative property the order does
not change. What is in parentheses does.
• Commutative Property of Addition
a+b=b+a
• Commutative Property of Multiplication
a•b=b•a
• In the commutative property, the order of
the problem does not matter.
• Identity Property of Addition
0+a=a
• Identity Property of Multiplication
1•a=a
1. 6 • 4 = 4 • 6
2. 5  1 = 5
3. (5 + 3) + 8 = 5 + (3 + 8)
4. (12 + 10) + 22 = 22 + (12 + 10)
5. 12 + 0 = 12
Finish the problem (Use the properties):
6. 5 + 3 =
7. (x + y) + z =
8. -22 + 0 =
9. 25  1 =
10. 2(4) =
1. 6 • 4 = 4 • 6 [ CPM ]
2. 5  1 = 5 [ IPM ]
3. (5 + 3) + 8 = 5 + (3 + 8) [ APA ]
4. (12 + 10) + 22 = 22 + (12 + 10) [ CPA ]
5. 12 + 0 = 12 [ IPA ]
Finish the problem (Use the properties):
6. 5 + 3 = 3 + 5 [ CPA ]
7. (x + y) + z = x + (y + z) [ APA ]
8. -22 + 0 = -22 [IPA]
9. 25  1 = 25 [IPM]
10. 2(4) = 4(2) [CPM]
Multiplying Monomials
• When two monomials have the same
variable, they can be multiplied
1. Multiply the numbers.
2. Add the exponents on the like variables.
Ex: 5y2  (-7y4)
= -35y6
Multiplying Monomials by
Polynomials
• –5t ( 2t2 – 7t + 9)
-5t (2t2) -5t (-7t) -5t (9)
-10t3 + 35t2 – 45t
• 3x3( 2x3 – 3x2 + 5x – 9)
3x3(2x3) 3x3(-3x2) 3x3(5x) 3x3(-9)
6x6 – 9x5 + 15x4 -27x3
Get your homework assignment out
and work these problems.
1.) x (3x – 5) + 2 (3x – 5)
2.) 3x (4x – 2) – 6 (4x – 2)
3.) 5x2 (2x + 6) – 3x (2x + 6)
Homework Answers
1.) 10t3 -16t2 + 33t
2.) -21y3 + 49y2 + 9y
3.) -4x3 + 15x2 – 2x
4.) 10b3 – 28b2 – 11b
5.) 3d3 + 4d2 – 35d
Multiplying Polynomials
1. Multiply the 2nd parentheses by the first
term in the 1st parentheses.
2. Multiply the 2nd parentheses by the second
term in the 1st parentheses.
3. Combine like terms.
Matrices
 Matrix – A rectangular arrangement of
numbers into rows and columns.
3
1
0
rows
-1
2
4
columns
 A matrix is identified by the number of rows and
columns.
 rows x columns
 The matrix on the previous slide has 2 rows and 3
columns so it is 2 x 3 matrix. Read “2 by 3”.
 The numbers in a matrix are called entries. In the
matrix on the previous slide, the entry in the second
row and third column is 4.
Adding & Subtracting Matrices
 You cannot add or subtract matrices that
have different numbers of rows and
columns.
 Use your addition rules and add the
corresponding entries and put them in the
same spot in your matrix answer.
Subtracting Matrices
1.) Change to addition. (First matrix stays the same,
subtraction sign becomes addition, EACH ENTRY IN
THE SECOND MATRIX MUST BE CHANGED TO
ITS OPPOSITE.
2.) Use your rule for adding matrices to get your answer.