Download PowerPoint Student

Document related concepts

History of logarithms wikipedia, lookup

Georg Cantor's first set theory article wikipedia, lookup

Infinity wikipedia, lookup

Law of large numbers wikipedia, lookup

Infinitesimal wikipedia, lookup

Positional notation wikipedia, lookup

Surreal number wikipedia, lookup

Fundamental theorem of algebra wikipedia, lookup

Large numbers wikipedia, lookup

Hyperreal number wikipedia, lookup

Proofs of Fermat's little theorem wikipedia, lookup

Real number wikipedia, lookup

Location arithmetic wikipedia, lookup

Algebra wikipedia, lookup

Mathematics of radio engineering wikipedia, lookup

Division by zero wikipedia, lookup

Arithmetic wikipedia, lookup

Elementary mathematics wikipedia, lookup

Addition wikipedia, lookup

Transcript
Algebra 1:
Topic 1 Notes
Table of Contents
1.
2.
3.
4.
5.
6.
7.
Order of Operations & Evaluating Expressions
The Real Number System
Properties of Real Numbers
Simplifying Radicals
Basic Exponent Properties
One & Two-Step Equations
Basic Multi-Step Equations
Order of Operations
& Evaluating
Expressions
Review: Order of Operations
Please- Parentheses
Excuse- Exponents
My- Multiplication
Dear- Division
Aunt- Addition
Sally- Subtraction
Review: Order of Operations
Remember that Multiplication and
Division are a pair…do in order from
left to right.
Addition and Subtraction are also a
pair….do in order from left to right.
Review: “Evaluate”
• Evaluating an expression means to plug in!
• For example:
2n + 4 for n = 3
2(3) + 4
10
Let’s Practice…
Evaluate the following expressions.
1. 3r – 17 for r = 6
2. 10 – 6d for d = 4
3. 5g + 20 – (4g) for g = 5
4. (3 + p)2 – 2p for p = 7
The Real Number
System:
Numbers Have
Names?!?!
Natural Numbers
• Non-decimal, positive numbers starting with
one.
Whole Numbers
• Non-decimal, positive numbers and zero.
Integers
• Non-decimal
positive and
negative numbers,
including zero.
Rational Numbers
• Any number that can be expressed as the
𝑝
quotient or fraction of two integers.
𝑞
• YES:
– Any integers
– Any decimals that
ends or repeats
– Any fraction
• NO:
– Never ending decimals
Irrational Numbers
• Any number that can not be expressed as a
fraction.
• Usually a never-ending, non-repeating
decimal.
• Examples:
𝜋
2, 5
1.2658945625692….
Let’s Practice…
Rational or Irrational.
1.
2
17
2.
3.
4.
22
3
27
1
3
Properties of
Real Numbers
List of Properties of Real
Numbers
•
•
•
•
•
Commutative
Associative
Distributive
Identity
Inverse
Commutative Properties
Definition: Changing the order of the
numbers in addition or multiplication will
not change the result.
Commutative Properties
Property
Addition
Multiplication
Numbers
Algebra
Associative Properties
Definition: Changing the
grouping of the numbers
in addition or
multiplication will not
change the result.
Associative Properties
Property
Addition
Multiplication
Numbers
Algebra
Distributive Property
Multiplication distributes over addition.
ab  c   ab  ac
32  5  3  2  3  5
Additive Identity Property
Definition: Zero preserves identities under
addition. In other words, adding zero to a
number does not change its value.
Example:
Multiplicative Identity
Property
Definition: The number 1 preserves
identities under multiplication. In other
words, multiplying a number by 1 does not
change the value of the number.
Example:
Additive Inverse Property
Definition: For each real number a there
exists a unique real number –a such that
their sum is zero. In other words, opposites
add to zero.
Example:
Multiplicative Inverse
Property
Definition: For each real number a there
1
exists a unique real number
1 𝑎 such that their
product is 1.
a
Example:
Simplifying
Radicals
Radical Vocab
How to Simplify Radicals
1. Make a factor tree of the radicand.
2. Circle all final factor pairs.
3. All circled pairs move outside the radical and
become single value.
4. Multiply all values outside radical.
5. Multiply all final factors that were not circled.
Place product under radical sign.
Let’s Practice…
1. 225
2. 300
Let’s Practice…
3.
1
49
4. 120
How to Simplify Cubed
Radicals
1. Make a factor tree of the radicand.
2. Circle all final factor groups of three.
3. All circled groups of three move outside the
radical and become single value.
4. Multiply all values outside radical.
5. Multiply all final factors that were not circled.
Place product under radical sign.
Let’s Practice…
3
1. 375
3
2. 64
Let’s Practice…
3
3. 81
3
4. 256
Exponents
Definition: Exponent
• The exponent of a number says how many
times to use that number in a multiplication.
It is written as a small number to the right
and above the base number.
The Zero Exponent Rule
• Any number (excluding zero) to the zero
power is always equal to one.
• Examples:
 1000=1
 1470=1
 550 =1
Negative Power Rule
Let’s Practice…
1. 5-2
2.
3 −2
4
3. (-3)-3
The One Exponent Rule
• Any number (excluding zero) to the first power
is always equal to that number.
• Examples:
 a1 = a
 71 = 7
 531 = 53
Simplifying Exponential
Expressions
Step 1: Simplify (get rid of negatives and
fractions, where possible).
Step 2: Plug in.
Let’s Practice…
8b3
4a−2
1. 4m3j-2
2.
for m = -2 and j = 3
for b = 2 and a = 3
Solving 2-step
Equations
Inverse Operations Pairs
Addition
Subtraction
Multiplication
Division
Exponents
Radicals
How to Solve 2-Step Equations
1. Use the opposite operation to move
the letters to the left and numbers to
the right.
2. Divide to get the variable alone.
Example
5x - 18 = 22
Let’s Practice…
1. 10x – 7 = 77
2. 15 – 3m = 45
Let’s Practice…
3. 4y = 3y + 12
4. 9b = 17 – 8b
Fractions…
To get rid of fractions, multiply all other
values by the reciprocal!
Example: k  6  2
4
Let’s Practice…
m
14

 24
1.
10
2. 3 −
2
x
5
=6
Solving
Multi-Step
Equations
How to solve Multi-Step Equations
1. Distribute.
Bonus Step: Multiply by reciprocal/LCM, if fractions.
2. Combine Like Terms (ONLY on same side of
equal sign).
3. Use the inverse operation to move numbers
to the right.
4. Use the inverse operation to move variables
to the left.
5. Divide.
Example
3(4x – 10) = 26
Example
6(4 + 3x) - 10x = 30 - 2x
Example
1
x – 8 = 4 + 2x
5
Let’s Practice…
4(5 + 2x) + 7x = -5
Let’s Practice…
3
𝑥
2
+6=−
7
2
+ 5𝑥