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Chabot Mathematics. §1.6 Exponent Properties. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected] MTH 55. 1.5. Review §. Any QUESTIONS About §1.5 → (Word) Problem Solving Any QUESTIONS About HomeWork §1.5 → HW-01. Exponent PRODUCT Rule.

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slide1

Chabot Mathematics

§1.6 ExponentProperties

Bruce Mayer, PE

Licensed Electrical & Mechanical [email protected]

review

MTH 55

1.5

Review §
  • Any QUESTIONS About
    • §1.5 → (Word) Problem Solving
  • Any QUESTIONS About HomeWork
    • §1.5 → HW-01
exponent product rule
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
example product rule
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)

example product rule1
Example  Product Rule
  • Solution a) x3x5 = x3+5Adding exponents

= x8

  • Solution b) 62 67  63 = 62+7+3

= 612

  • Solution c) (x + y)6(x + y)9 = (x + y)6+9

= (x + y)15

  • Solution d) (w3z4)(w3z7) = w3z4w3z7

= w3w3z4z7

= w6z11

Base is x

Base is 6

Base is (x + y)

TWO Bases: w & z

exponent quotient rule
Exponent QUOTIENT Rule
  • For any nonzero number a and any positive integers m & n for which m > n,
  • In other Words: To DIVIDE powers with the same base, SUBTRACT the exponent of the denominator from the exponent of the numerator
example quotient rule
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)
example quotient rule1
Example  Quotient Rule
  • Solution a)

Base is x

  • Solution b)

Base is 8

  • Solution c)

Base is (6y)

  • Solution d)

TWO Bases: r & t

the exponent zero
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
example the exponent zero
Example  The Exponent Zero
  • Simplify: a) 12450 b) (−3)0c) (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
the power rule
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
example power rule
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

raising a product to a power
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
example product to power
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

= a14b21a4b5Multiplying exponents

= a18b26 Adding exponents

raising a quotient to a power
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
example quotient to a power
Example  Quotient to a Power
  • Simplify: a) b) c)
  • Solution a)
  • Solution b)
  • Solution c)
negative exponents
Negative Exponents
  • Integers as Negative Exponents
negative exponents1
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
example negative exponents
Example  Negative Exponents
  • Express using POSITIVE exponents, and, if possible, simplify.

a) m–5b) 5–2 c) (−4)−2 d) xy–1

  • SOLUTION

a) m–5 =

b) 5–2 =

example negative exponents1
Example  Negative Exponents
  • Express using POSITIVE exponents, and, if possible, simplify.

a) m–5 b) 5–2c) (−4)−2d) xy−1

  • SOLUTION

c) (−4)−2=

d) xy–1 =

  • Remember PEMDAS
more examples
More Examples
  • Simplify. Do NOT use NEGATIVE exponents in the answer.a) b) (x4)3 c) (3a2b4)3d) e) f)
  • Solution

a)

more examples1
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)

factors negative exponents
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
examples flippers
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.
reciprocals negative exponents
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
examples flippers1
Examples  Flippers
  • Simplify
  • SOLUTION
summary exponent properties
Summary – Exponent Properties

This summary assumes that no denominators are 0 and that 00 is not considered. For any integers m and n

whiteboard work
WhiteBoard Work
  • Problems From §1.6 Exercise Set
    • 14, 24, 52, 70, 84, 92, 112, 130
  • Base & Exponent →Which is Which?
all done for today
All Done for Today

AstronomicalUnit

(AU)

slide35

Chabot Mathematics

Appendix

Bruce Mayer, PE

Licensed Electrical & Mechanical [email protected]

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