Show that (-18),6, (-7) hold associative property for multiplication
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Answer:
There are many times in algebra when you need to simplify an expression. The properties of real numbers provide tools to help you take a complicated expression and simplify it.
The associative, commutative, and distributive properties of algebra are the properties most often used to simplify algebraic expressions. You will want to have a good understanding of these properties to make the problems in algebra easier to work.
The Commutative Properties of Addition and Multiplication
You may encounter daily routines in which the order of tasks can be switched without changing the outcome. For example, think of pouring a cup of coffee in the morning. You would end up with the same tasty cup of coffee whether you added the ingredients in either of the following ways:
· Pour 12 ounces of coffee into mug, then add splash of milk.
· Add a splash of milk to mug, then add 12 ounces of coffee.
The order that you add ingredients does not matter. In the same way, it does not matter whether you put on your left shoe or right shoe first before heading out to work. As long as you are wearing both shoes when you leave your house, you are on the right track!
In mathematics, we say that these situations are commutative—the outcome will be the same (the coffee is prepared to your liking; you leave the house with both shoes on) no matter the order in which the tasks are done.
Likewise, the commutative property of addition states that when two numbers are being added, their order can be changed without affecting the sum. For example, 30 + 25 has the same sum as 25 + 30.
30 + 25 = 55
25 + 30 = 55
Multiplication behaves in a similar way. The commutative property of multiplication states that when two numbers are being multiplied, their order can be changed without affecting the product. For example, 7 · 12 has the same product as 12 · 7.
7 · 12 = 84
12 · 7 = 84
These properties apply to all real numbers. Let’s take a look at a few addition examples.
Original Equation
Rewritten Equation
1.2 + 3.8 = 5
3.8 + 1.2 = 5
14 + (−10) = 4
(−10) + 14 = 4
(−5.2) + (−3.6) = −8.8
(−3.6) + (−5.2) = −8.8
Commutative Property of Addition
For any real numbers a and b, a + b = b + a.
Subtraction is not commutative. For example, 4 − 7 does not have the same difference as 7 − 4. The − sign here means subtraction.
However, recall that 4 − 7 can be rewritten as 4 + (−7), since subtracting a number is the same as adding its opposite. Applying the commutative property for addition here, you can say that 4 + (−7) is the same as (−7) + 4. Notice how this expression is very different than 7 – 4.
Now look at some multiplication examples.
Original Equation
Rewritten Equation
4.5 · 2 = 9
2 · 4.5 = 9
(−5) · 3 = -15
3 · (−5) = -15
Commutative Property of Multiplication
For any real numbers a and b, a · b = b · a.
Order does not matter as long as the two quantities are being multiplied together. This property works for real numbers and for variables that represent real numbers.
Just as subtraction is not commutative, neither is division commutative. 4 ÷ 2 does not have the same quotient as 2 ÷ 4.
Example
Problem
Write the expression (−15.5) + 35.5 in a different way, using the commutative property of addition, and show that both expressions result in the same answer.
(−15.5) + 35.5 = 20
35.5 + (−15.5)
35.5 + (−15.5)
35.5 – 15.5 = 20
Adding.
Using the commutative property, you can switch the −15.5 and the 35.5 so that they are in a different order.
Adding 35.5 and −15.5 is the same as subtracting 15.5 from 35.5. The sum is 20.
Answer (−15.5) + 35.5 = 20 and 35.5 + (−15.5) = 20
Rewrite 52 • y in a different way, using the commutative property of multiplication. Note that y represents a real number.
A) 5y • 2
B) 52y
C) 26 • 2 • y
D) y • 52
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