Properties of Exponents

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Properties of Exponents
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Definition of Exponent

• An exponent is the power p in an expression ap.
• 52
• The number 5 is the base.
• The number 2 is the exponent.
• The exponent is an instruction that tells us how
  many times to use the base in a multiplication.

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Examples

• 43
• -34
• (-2)5
• (-3/4)2

• (4)(4)(4) 64
• (-)(3)(3)(3)(3) -81
• (-2)(-2)(-2)(-2)(-2) -32
• (-3/4)(-3/4) (9/16)

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Puzzler

• (-5)2 -52?
• (-5)(-5) -(5)(5)
• 25 -25

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Multiplication with Exponents by Definition

• 32 35
• (3)(3) (3)(3)(3)(3)(3)
• 37
• Note 257

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Property 1 for Exponents

• If a is any real number and r and s are integers,
  then

To multiply two expressions with the same base,
add exponents and use the common base.
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Examples of Property 1
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Examples of Property 1
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Examples of Property 1
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Power to a Power by Definition

• (32)3
• ((3)(3))1((3)(3))1((3)(3))1
• 36
• Note 3(2)6

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Property 2 for Exponents

• If a is any real number and r and s are integers,
  then

A power raised to another power is the base
raised to the product of the powers.
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Examples of Property 2
One base, two exponents multiply the exponents.
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Property 3 for Exponents

• If a and b are any real number and r is an
  integer, then

Distribute the exponent.
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Examples of Property 3
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Examples of Property 3
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By the Definition of Exponents
Notice that 5 3 2
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Examples
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Negative Exponents
Notice that 3 5 -2
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Definition of Negative Exponents

• If r is a positive integer, then

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Examples of Negative Exponents
Notice that Negative Exponents do not indicate
negative numbers.
Negative exponents do indicate Reciprocals.
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Examples of Negative Exponents
Notice that exponent does not touch the 3.
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Property 4 for Exponents

• If a is any real number and r and s are integers,
  then

To divide like bases subtract the exponents.
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Property 5 for Exponents

• If a and b are any two real numbers and r is an
  integer, then

The power of a quotient is the quotient of the
powers.
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Property 5 Example
The power of a quotient is the quotient of the
powers.
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Zero as an Exponent
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Zero to the Zero?
Undefined
STOP
Zeros are not allowed in the denominator. So 00
is undefined.
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Examples
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Exponent Rules Summary
Properties
Definitions
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Definition Scientific Notation

• A number is in scientific notation when it is
  written as the product of a number between 1 and
  10 and an integer power of 10.
• A number written in scientific notation has the
  form
• where 1lt n lt 10 and r an integer.

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Write 1349.7 in scientific notation.
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Write 1.85 x 104 in expanded notation.
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More on Scientific Notation
How far does the decimal point move?