Bruce Mayer, PE Licensed Electrical &amp; Mechanical Engineer BMayer@ChabotCollege

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Chabot Mathematics. §1.6 Exponent Properties. Bruce Mayer, PE Licensed Electrical &amp; Mechanical Engineer BMayer@ChabotCollege.edu. 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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Chabot Mathematics

§1.6 ExponentProperties

Bruce Mayer, PE

MTH 55

1.5

Review §
• §1.5 → (Word) Problem Solving
• §1.5 → HW-01
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
• 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 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
• 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
• 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 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
• 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
• 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
• 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
• 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
• 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
• 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

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
• Simplify: a) b) c)
• Solution a)
• Solution b)
• Solution c)
Negative Exponents
• Integers as Negative Exponents
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
• 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 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
• Simplify. Do NOT use NEGATIVE exponents in the answer.a) b) (x4)3 c) (3a2b4)3d) e) f)
• Solution

a)

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
• 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
• 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
• 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  Flippers
• Simplify
• SOLUTION
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
• Problems From §1.6 Exercise Set
• 14, 24, 52, 70, 84, 92, 112, 130
• Base & Exponent →Which is Which?
All Done for Today

AstronomicalUnit

(AU)

Chabot Mathematics

Appendix

Bruce Mayer, PE