Exponential Notation.

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Section 1.4

• Exponential Notation.
• Order of Operations.

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Exponential Notation

• Like multiplication which was originally
  introduced as a shortcut for repetitive addition,
  exponential notation was originally introduced as
  a shortcut for repetitive multiplication
• The number 2 that is multiplied is called a base,
  12 the number of times the base is multiplied is
  called an exponent.

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Definition
Exponential Notation
If number b is multiplied by itself n times, this product could be shortly written as This form is called exponential notation. The number b that is multiplied is called a base, n the number of times the base is multiplied is called an exponent.
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Example

• Rewrite in exponential notation
• Evaluate exponential expression by repetitive
  multiplication

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Example

• Evaluate an expression
• If we evaluate addition first and then
  multiplication, we will get
• If we evaluate multiplication first and then
  addition, we will get
•  
• In the example above the result depended on the
  order in which we execute different operations.
• When evaluating expressions with more than one
  operation we need to know which operation should
  be done first, which next. In other words we need
  to know the order of operations.

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The Order of Operations

1. If there are any grouping symbols, such as
   parenthesis ( ), square brackets , curly
   braces or the other grouping symbols perform
   all the operations within the grouping symbols
   first.
2. Evaluate all the exponential expressions.
3. Do all negations put all negative numbers in the
   parenthesis.
4. Do all multiplications and divisions from left to
   right.
5. Do all additions and subtractions from left to
   right.

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Example

• Evaluate an expression

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Properties of Algebraic Operations
Addition of any number to 0 or 0 to any number always results in 0 an original number. Multiplication of any number by 0 or 0 by any number always results in 0 Multiplication of any number by 1 or 1 by any number always results in an original number
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Commutative Property

• Of Addition
• The sum of any two numbers a and b does not
  depend on the order in which they are added
• Of Multiplication
• The product of any two numbers a and b does not
  depend on the order in which they are multiplied

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Associative Property

• Of Addition
• The sum of any three numbers a,b and c does could
  be found by grouping the numbers in any order of
  addition
• Of Multiplication
• The product of any three numbers a,b and c does
  could be found by grouping the numbers in any
  order of multiplication

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Distributive Property

• If an expression in parenthesis is multiplied by
  a factor, to open the parenthesis, multiply each
  term inside of the parenthesis by this factor