Simplifying Expressions Using the Commutative and Associative Properties
Simplifying Expressions Using the Commutative and Associative Properties
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 part (b) of the first example in the previous lesson was easier to simplify than part (a) because the opposites were next to each other and their sum is \(0.\) Likewise, part (b) in the second example was easier, with the reciprocals grouped together, because their product is \(1.\) In the next few examples, we’ll use our number sense to look for ways to apply these properties to make our work easier.
Example
Simplify: \(-84n+\left(-73n\right)+84n.\)
Solution
Notice the first and third terms are opposites, so we can use the commutative property of addition to reorder the terms.
| \(-84n+\left(-73n\right)+84n\) | |
| Re-order the terms. | \(-84n+84n+\left(-73n\right)\) |
| Add left to right. | \(0+\left(-73n\right)\) |
| Add. | \(-73n\) |
Now we will see how recognizing reciprocals is helpful. Before multiplying left to right, look for reciprocals—their product is \(1.\)
Example
Simplify: \(\frac{7}{15}·\frac{8}{23}·\frac{15}{7}.\)
Solution
Notice the first and third terms are reciprocals, so we can use the Commutative Property of Multiplication to reorder the factors.
| \(\frac{7}{15}·\frac{8}{23}·\frac{15}{7}\) | |
| Re-order the terms. | \(\frac{7}{15}·\frac{15}{7}·\frac{8}{23}\) |
| Multiply left to right. | \(1·\frac{8}{23}\) |
| Multiply. | \(\frac{8}{23}\) |
In expressions where we need to add or subtract three or more fractions, combine those with a common denominator first.
Example
Simplify: \(\left(\frac{5}{13}+\frac{3}{4}\right)+\frac{1}{4}.\)
Solution
Notice that the second and third terms have a common denominator, so this work will be easier if we change the grouping.
| \(\left(\frac{5}{13}+\frac{3}{4}\right)+\frac{1}{4}\) | |
| Group the terms with a common denominator. | \(\frac{5}{13}+\left(\frac{3}{4}+\frac{1}{4}\right)\) |
| Add in the parentheses first. | \(\frac{5}{13}+\left(\frac{4}{4}\right)\) |
| Simplify the fraction. | \(\frac{5}{13}+1\) |
| Add. | \(1\frac{5}{13}\) |
| Convert to an improper fraction. | \(\frac{18}{13}\) |
When adding and subtracting three or more terms involving decimals, look for terms that combine to give whole numbers.
Example
Simplify: \(\left(6.47q+9.99q\right)+1.01q.\)
Solution
Notice that the sum of the second and third coefficients is a whole number.
| \(\left(6.47q+9.99q\right)+1.01q\) | |
| Change the grouping. | \(6.47q+\left(9.99q+1.01q\right)\) |
| Add in the parentheses first. | \(6.47q+\left(11.00q\right)\) |
| Add. | \(17.47q\) |
Many people have good number sense when they deal with money. Think about adding \(99\) cents and \(1\) cent. Do you see how this applies to adding \(9.99+1.01?\)
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.
Example
Simplify the expression: \(\left[1.67\left(8\right)\right]\left(0.25\right).\)
Solution
Notice that multiplying \(\left(8\right)\left(0.25\right)\) is easier than multiplying \(1.67\left(8\right)\) because it gives a whole number. (Think about having \(8\) quarters—that makes \(\text{\$2.)}\)
| \(\left[1.67\left(8\right)\right]\left(0.25\right)\) | |
| Regroup. | \(1.67\left[\left(8\right)\left(0.25\right)\right]\) |
| Multiply in the brackets first. | \(1.67\left[2\right]\) |
| Multiply. | \(3.34\) |
When simplifying expressions that contain variables, we can use the commutative and associative properties to re-order or regroup terms, as shown in the next pair of examples.
Example
Simplify: \(6\left(9x\right).\)
Solution
| \(6\left(9x\right)\) | |
| Use the associative property of multiplication to re-group. | \(\left(6·9\right)x\) |
| Multiply in the parentheses. | \(54x\) |
In Mathematics 102, we learned to combine like terms by rearranging an expression so the like terms were together. We simplified the expression \(3x+7+4x+5\) by rewriting it as \(3x+4x+7+5\) and then simplified it to \(7x+12.\) We were using the Commutative Property of Addition.
Example
Simplify: \(18p+6q+\left(-15p\right)+5q.\)
Solution
Use the Commutative Property of Addition to re-order so that like terms are together.
| \(18p+6q+\left(-15p\right)+5q\) | |
| Re-order terms. | \(18p+\left(-15p\right)+6q+5q\) |
| Combine like terms. | \(3p+11q\) |
This lesson is part of:
Properties of Real Numbers