The Associative and Commutative Rules for Multiplication
The Associative Rule for Multiplication
Both addition and multiplication satisfy the associative rule. It states that, when adding or when multiplying, you can arrive at the correct sum or product no matter how you group your numbers. To effectively apply this rule, parentheses are used to group the numbers to demonstrate which numbers are added/multiplied first. The associative rule for multiplication can be justified by considering the volume of boxes (i.e. rectangular parallelepipeds).
The Associative Rule for Multiplication: For non-negative integers, a, b, and c
a × (b × c) = (a × b) × c
Example #8:
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The Associative Rule for Multiplication: (5 × 7 × 4) let a, b, c be defined as a = 5, b = 7, c = 4 |
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5 × (7 × 4) = 5 × (28) = 140 |
(5 × 7) × 4 = (35) × 4 = 140 |
The Law of Exponents: for any whole numbers a and b, we have the equality
xa • xb = xa+b
The argument for this is similar to example 9b, although more complicated.
Example #9a:
The Associative Rule for Multiplication:
(x • x2)
(x • x2) = x (x • x) = (x • x) x = x2 x = x3
Example #9b:
(x • x3)
(x • x3) = x(x • x • x) = (x • x) • (x • x) = x2 x2 = x2 •(x • x)
= (x2 • x) • x = x3 • x = x4
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