#set document(title: "14.1 Properties of Numbers", author: "OpenStax / XYZ Homework")
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== 14.1#h(0.6em)Properties of Numbers


    #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Associative Laws]


      - #strong[Addition:]  #linebreak() 
			If #math.equation(block: false, alt: "a")[$a$], #math.equation(block: false, alt: "b")[$b$], and #math.equation(block: false, alt: "c")[$c$] are any numbers, then #math.equation(block: false, alt: "open parenthesis a plus b close parenthesis plus c equals a plus open parenthesis b plus c close parenthesis")[$( a + b ) + c = a + ( b + c )$].
- #strong[Multiplication]  #linebreak() 
			If #math.equation(block: false, alt: "a")[$a$], #math.equation(block: false, alt: "b")[$b$], and #math.equation(block: false, alt: "c")[$c$] are any numbers, then #math.equation(block: false, alt: "open parenthesis a times b close parenthesis times c equals a times open parenthesis b times c close parenthesis")[$( a ⋅ b ) ⋅ c = a ⋅ ( b ⋅ c )$].
]


    #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Commutative Laws]


      - #strong[Addition:]  #linebreak() 
			If #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$] are any numbers, then #math.equation(block: false, alt: "a plus b equals b plus a")[$a + b = b + a$].
- #strong[Multiplication]  #linebreak() 
			If #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$] are any numbers, then #math.equation(block: false, alt: "a times b equals b times a")[$a ⋅ b = b ⋅ a$].
]


    #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Distributive Law]


      #math.equation(block: false, alt: "a open parenthesis b plus c close parenthesis equals a b plus a c")[$a ( b + c ) = a b + a c$] for any numbers #math.equation(block: false, alt: "a")[$a$], #math.equation(block: false, alt: "b")[$b$], and #math.equation(block: false, alt: "c")[$c$].
]


    #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Properties of Equality]


      - #strong[Addition:]  #linebreak() 
			If #math.equation(block: false, alt: "a equals b")[$a = b$] and #math.equation(block: false, alt: "c")[$c$] is any number, then #math.equation(block: false, alt: "a plus c equals b plus c")[$a + c = b + c$].
- #strong[Subtraction:]  #linebreak() 
			If #math.equation(block: false, alt: "a equals b")[$a = b$] and #math.equation(block: false, alt: "c")[$c$] is any number, then #math.equation(block: false, alt: "a minus c equals b minus c")[$a − c = b − c$].
- #strong[Multiplication]  #linebreak() 
			If #math.equation(block: false, alt: "a equals b")[$a = b$] and #math.equation(block: false, alt: "c")[$c$] is any number, then #math.equation(block: false, alt: "a times c equals b times c")[$a ⋅ c = b ⋅ c$].
- #strong[Division]  #linebreak() 
			If #math.equation(block: false, alt: "a equals b")[$a = b$] and #math.equation(block: false, alt: "c")[$c$] is any nonzero number, then #math.equation(block: false, alt: "the fraction a over c equals the fraction b over c")[$frac(a, c) = frac(b, c)$].
]


    #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Fundamental Principle of Fractions]


      If #math.equation(block: false, alt: "a")[$a$] is any number, and #math.equation(block: false, alt: "b")[$b$] and #math.equation(block: false, alt: "c")[$c$] are nonzero numbers, then #math.equation(block: false, alt: "the fraction a times c over b times c equals the fraction a over b")[$display(frac(a ⋅ c, b ⋅ c) = frac(a, b))$].
]


    #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Laws of Exponents]


      + #math.equation(block: false, alt: "a to the power m times a to the power n equals a to the power m plus n")[$a^(m) ⋅ a^(n) = a^(m + n)$]
+ - #math.equation(block: false, alt: "the fraction a to the power m over a to the power n equals a to the power m minus n open parenthesis n less than m close parenthesis")[$display(frac(a^(m), a^(n))) = a^(m − n) #hide($b l a n k 1$) ( n < m )$]
- #math.equation(block: false, alt: "the fraction a to the power m over a to the power n equals the fraction 1 over a to the power n minus m open parenthesis n greater than m close parenthesis")[$display(frac(a^(m), a^(n)) = frac(1, a^(n − m)) #hide($b l a n k$) ( n > m ))$]
+ #math.equation(block: false, alt: "open parenthesis a to the power m close parenthesis to the power n equals a to the power m plus n")[$attach(( a^(m) ), t: n) = a^(m + n)$]
+ #math.equation(block: false, alt: "open parenthesis a b close parenthesis to the power n equals a to the power n b to the power n")[$( a b )^(n) = a^(n) b^(n)$]
+ #math.equation(block: false, alt: "open parenthesis the fraction a over b close parenthesis to the power n equals the fraction a to the power n over b to the power n")[$display(attach(( frac(a, b) ), t: n) = frac(a^(n), b^(n)))$]
]


    #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Product Rule for Radicals]


      If #math.equation(block: false, alt: "a")[$a$]  and #math.equation(block: false, alt: "b")[$b$] are both nonnegative, then #math.equation(block: false, alt: "the square root of a b equals the square root of a the square root of b")[$sqrt(a b) = sqrt(a) sqrt(b)$].
]


    #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Quotient Rule for Radicals]


      If #math.equation(block: false, alt: "a greater than or equal to 0")[$a ≥ 0$]  and #math.equation(block: false, alt: "b greater than 0")[$b > 0$], then #math.equation(block: false, alt: "the square root of the fraction a over b equals the fraction the square root of a over the square root of b")[$sqrt(display(frac(a, b))) = display(frac(sqrt(a), sqrt(b)))$].
]


    #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Zero-Factor Principle]


      If #math.equation(block: false, alt: "a b equals 0")[$a b = 0$]  then either #math.equation(block: false, alt: "a equals 0")[$a = 0$] or #math.equation(block: false, alt: "b equals 0")[$b = 0$].
]


    #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Properties of Absolute Value]


      #math.equation(block: true, alt: "vertical bar a plus b vertical bar less than or equal to vertical bar a vertical bar plus vertical bar b vertical bar, Triangle inequality; vertical bar a b vertical bar equals vertical bar a vertical bar vertical bar b vertical bar, Multiplicative property")[$| a + b | ≤ | a | + | b | &  & "Triangle inequality" \ | a b | = | a | | b | &  & "Multiplicative property "$]
]