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Evaluate each expression when
ⓐ
ⓑ
| ⓐ | |
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| Substitute 17 for n. |
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| Multiply in the parentheses first. |
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| Multiply again. |
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| ⓑ | |
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| Substitute 17 for n. |
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| Multiply. The product of reciprocals is 1. |
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| Multiply again. |
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What was the difference between part ⓐ and part ⓑ here? Only the grouping changed. By the Associative Property of Multiplication, By carefully choosing how to group the factors, we can make the work easier.
When we have to simplify algebraic expressions, we can often make the work easier by applying the Commutative or Associative Property first instead of automatically following the order of operations. Notice that in [link] part ⓑ was easier to simplify than part ⓐ because the opposites were next to each other and their sum is Likewise, part ⓑ in [link] was easier, with the reciprocals grouped together, because their product is In the next few examples, we’ll use our number sense to look for ways to apply these properties to make our work easier.
Simplify:
Notice the first and third terms are opposites, so we can use the commutative property of addition to reorder the terms.
| Re-order the terms. | |
| Add left to right. | |
| Add. |
Now we will see how recognizing reciprocals is helpful. Before multiplying left to right, look for reciprocals—their product is
Simplify:
Notice the first and third terms are reciprocals, so we can use the Commutative Property of Multiplication to reorder the factors.
| Re-order the terms. | |
| Add left to right. | |
| Add. |
In expressions where we need to add or subtract three or more fractions, combine those with a common denominator first.
Simplify:
Notice that the second and third terms have a common denominator, so this work will be easier if we change the grouping.
| Group the terms with a common denominator. | |
| Add in the parentheses first. | |
| Simplify the fraction. | |
| Add. | |
| Convert to an improper fraction. |
When adding and subtracting three or more terms involving decimals, look for terms that combine to give whole numbers.
Simplify:
Notice that the sum of the second and third coefficients is a whole number.
| Change the grouping. | |
| Add in the parentheses first. | |
| Add. |
Many people have good number sense when they deal with money. Think about adding cents and cent. Do you see how this applies to adding
No matter what you are doing, it is always a good idea to think ahead. When simplifying an expression, think about what your steps will be. The next example will show you how using the Associative Property of Multiplication can make your work easier if you plan ahead.
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