W A T K I N S - J O H N S O N C O M P A N Y Semiconductor Equipment Group

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Chabot Mathematics
1.6 ExponentProperties
Bruce Mayer, PE Licensed Electrical Mechanical
EngineerBMayer_at_ChabotCollege.edu
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Review

• Any QUESTIONS About
• 1.5 ? (Word) Problem Solving
• Any QUESTIONS About HomeWork
• 1.5 ? HW-01

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Exponent PRODUCT Rule

• For any number a and any positive integers m and
  n,

Exponent
Base

• In other Words To MULTIPLY powers with the same
  base, keep the base and ADD the exponents

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Quick Test of Product Rule

• Test

?
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Example ? Product Rule

• Multiply and simplify each of the following.
  (Here simplify means express the product as one
  base to a power whenever possible.)a) x3 ?
  x5 b) 62 ? 67 ? 63
• c) (x y)6(x y)9 d) (w3z4)(w3z7)

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Example ? Product Rule

• Solution a) x3 ? x5 x35 Adding exponents
• x8
• Solution b) 62 ? 67 ? 63 6273
• 612
• Solution c) (x y)6(x y)9 (x y)69
• (x y)15
• Solution d) (w3z4)(w3z7) w3z4w3z7
• w3w3z4z7
• w6z11

Base is x
Base is 6
Base is (x y)
TWO Bases w z
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Exponent QUOTIENT Rule

• For any nonzero number a and any positive
  integers m n for which m gt n,

• In other Words To DIVIDE powers with the same
  base, SUBTRACT the exponent of the denominator
  from the exponent of the numerator

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Quick Test of Quotient Rule

• Test

?
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Example ? Quotient Rule

• Divide and simplify each of the following. (Here
  simplify means express the product as one base
  to a power whenever possible.)
• a) b)

• c) d)

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Example ? Quotient Rule

• Solution a)

Base is x

• Solution b)

Base is 8

• Solution c)

Base is (6y)

• Solution d)

TWO Bases r t
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The Exponent Zero

• For any number a where a ? 0

• In other Words Any nonzero number raised to
  the 0 power is 1
• Remember the base can be ANY Number
• 0.00073, 19.19, -86, 1000000, anything

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Example ? The Exponent Zero

• Simplify a) 12450 b) (-3)0 c) (4w)0 d)
  (-1)80 e) -80
• Solutions
• 12450 1
• (-3)0 1
• (4w)0 1, for any w ? 0.
• (-1)80 (-1)1 -1
• -80 is read the opposite of 80 and is
  equivalent to (-1)80 -80 (-1)80 (-1)1 -1

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The POWER Rule

• For any number a and any whole numbers m and n

• In other Words To RAISE a POWER to a POWER,
  MULTIPLY the exponents and leave the base
  unchanged

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Quick Test of Power Rule

• Test

?
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Example ? Power Rule

• Simplify a) (x3)4 b) (42)8
• Solution a) (x3)4 x3?4
• x12
• Solution b) (42)8 42?8
• 416

Base is x
Base is 4
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Raising a Product to a Power

• For any numbers a and b and any whole number n,

• In other Words To RAISE A PRODUCT to a POWER,
  RAISE Each Factor to that POWER

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Quick Test of Product to Power

• Test

?
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Example ? Product to Power

• Simplify a) (3x)4 b) (-2x3)2
  c) (a2b3)7(a4b5)
• Solutions
• (3x)4 34x4 81x4
• (-2x3)2 (-2)2(x3)2 (-1)2(2)2(x3)2 4x6
• (a2b3)7(a4b5) (a2)7(b3)7a4b5
• a14b21a4b5 Multiplying exponents
• a18b26 Adding exponents

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Raising a Quotient to a Power

• For any real numbers a and b, b ? 0, and any
  whole number n

• In other Words To Raise a Quotient to a power,
  raise BOTH the numerator denominator to the
  power

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Quick Test of Quotient to Power

• Test

?
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Example ? Quotient to a Power

• Simplify a) b) c)

• Solution a)

• Solution b)

• Solution c)

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Negative Exponents

• Integers as Negative Exponents

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Negative Exponents

• For any real number a that is nonzero and any
  integer n

• The numbers a-n and an are thus RECIPROCALS of
  each other

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Example ? Negative Exponents

• Express using POSITIVE exponents, and, if
  possible, simplify.
• a) m5 b) 52 c) (-4)-2 d) xy1
• SOLUTION
• a) m5

• b) 52

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Example ? Negative Exponents

• Express using POSITIVE exponents, and, if
  possible, simplify.
• a) m5 b) 52 c) (-4)-2 d) xy-1
• SOLUTION
• c) (-4)-2

d) xy1

• Remember PEMDAS

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More Examples

• Simplify. Do NOT use NEGATIVE exponents in the
  answer.a) b) (x?4)?3 c) (3a2b?4)3d) e)
  f)

• Solution
• a)

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More Examples

• Solution
• b) (x-4)-3 x(-4)(-3) x12
• c) (3a2b-4)3 33(a2)3(b-4)3
• 27 a6b-12

• d)

• e)

• f)

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Factors Negative Exponents

• For any nonzero real numbers a and b and any
  integers m and n

• A factor can be moved to the other side of the
  fraction bar if the sign of the exponent is
  changed

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Examples ? Flippers

• Simplify

• SOLUTION
• We can move the negative factors to the other
  side of the fraction bar if we change the sign of
  each exponent.

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Reciprocals Negative Exponents

• For any nonzero real numbers a and b and any
  integer n

• Any base to a power is equal to the reciprocal of
  the base raised to the opposite power

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Examples ? Flippers

• Simplify

• SOLUTION

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Summary Exponent Properties
1 as an exponent a1 a
0 as an exponent a0 1
Negative Exponents(flippers)
The Product Rule
The Quotient Rule
The Power Rule (am)n amn
The Product to a Power Rule (ab)n anbn
The Quotient to a Power Rule
This summary assumes that no denominators are 0
and that 00 is not considered. For any integers m
and n
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WhiteBoard Work

• Problems From 1.6 Exercise Set
• 14, 24, 52, 70, 84, 92, 112, 130

• Base Exponent ?Which is Which?

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All Done for Today
AstronomicalUnit (AU)
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Chabot Mathematics
Appendix
Bruce Mayer, PE Licensed Electrical Mechanical
EngineerBMayer_at_ChabotCollege.edu