Transcript Just the facts: Order of Operations and Properties of real numbers

Algebra II
Chapter 2
2012
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Parenthesis – anything grouped… including
information above or below a fraction bar.
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Exponents – anything in the same family as a ‘power’…
this includes radicals (square roots).
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Multiplication- this includes distributive property
(discussed in detail later).
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Some items are grouped!!!
Multiplication and Division are GROUPED from left to
right (like reading a book- do whichever comes first.
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Addition and Subtraction are also grouped from left to
right, do whichever comes first in the problem.
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Parenthesis
Exponents
Multiplication and Division
Addition and Subtraction
In order from left to right
In order from left to right
16  4(3  1)  22 11 
3
16  4(2)  22 11 
3
Parenthesis
Exponents
16  4(8)  22  11 
4(8)  22  11 
This one is tricky!
Remember: Multiplication/Division are grouped from left to right…what comes 1st?
Division did…now do the multiplication (indicated by parenthesis)
32  22 11 
32  2 
More division
Subtraction
30
3(5)  65
3(2  3)  65
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2
2
2
2
Exponents
Parenthesis
75  65 10
3(25)  65
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2
2
2
Remember the division
symbol here is grouping
everything on top, so
work everything up there
first….multiplication
Subtraction
Division –
because all the
work is done
above and
below the line
5
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Parenthesis
Exponents
Multiplication
Division
Addition
Subtraction
1.
Can you fill in the missing operations?
2 - (3+5) + 4 = -2
2.
4 + 7 * 3 ÷ 3 = 11
3.
5 * 3 + 5 ÷ 2 = 10
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 Associative
Properties
 Commutative Properties
 Inverse Properties
 Identity Properties
 Distributive Property
All of these rules apply to Addition and Multiplication
It doesn’t matter how you group (associate) addition
or multiplication…the answer will be the same!
Rules:
Associative Property of Addition
(a+b)+c = a+(b+c)
Associative Property of
Multiplication
(ab)c = a(bc)
Samples:
Associative Property of Addition
(1+2)+3 = 1+(2+3)
Associative Property of Multiplication
(2x3)4 = 2(3x4)
It doesn’t matter how you swap addition or
multiplication around…the answer will be the same!
Rules:
Commutative Property of Addition
a+b = b+a
Commutative Property of
Multiplication
ab = ba
Samples:
Commutative Property of Addition
1+2 = 2+1
Commutative Property of Multiplication
(2x3) = (3x2)
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Does the Associative Property hold true for
Subtraction and Division?
Is (5-2)-3 = 5-(2-3)?
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Is (6/3)-2 the same as 6/(3-2)?
Does the Commutative Property hold true for
Subtraction and Division?
Is 5-2 = 2-5?
Is 6/3 the same as 3/6?
Properties of real numbers are only for Addition and Multiplication
What is the opposite (inverse) of addition?
What is the opposite of multiplication?
Rules:
Inverse Property of Addition
a+(-a) = 0
Inverse Property of
Multiplication
a(1/a) = 1
Subtraction (add the negative)
Division (multiply by reciprocal)
Samples:
Inverse Property of Addition
3+(-3)=0
Inverse Property of Multiplication
2(1/2)=1
What can you add to a number & get the same number back? 0 (zero)
What can you multiply a number by and get the number back? 1 (one)
Rules:
Identity Property of Addition
a+0 = a
Identity Property of
Multiplication
a(1) = a
Samples:
Identity Property of Addition
3+0=3
Identity Property of Multiplication
2(1)=2
If something is sitting just outside a set of parenthesis, you can
distribute it through the parenthesis with multiplication and
remove the parenthesis.
Rule:
a(b+c) = ab+bc
Samples:
4(3+2)=4(3)+4(2)=12+8=20
• 2(x+3) = 2x + 6
• -(3+x) = -3 - x