Q.
A. Combine what’s in the absolute value and parentheses first, before combining the results:
When you get to the three terms with subtract and add, 1 – 11 + 18, you always perform the operations in order, reading from left to right. (See Chapter 7 for more on this process, called the order of operations.)
Q.
A. You have to complete the work in the denominator first before dividing the 32 by that result:
Practice Questions
1. 3(2 – 5) + 14 =
2. 4[3(6 – 8) + 2(5 + 9)] – 11 =
3. 5{8[2 + (6 – 3)] – 4} =
4.
5.
6.
Practice Answers
1. 5.
2. 77.
3. 180.
4.
5. –56.
6. 8.
The distributive property is used to perform an operation on each of the terms within a grouping symbol. The following rules show distributing multiplication over addition and distributing multiplication over subtraction:
Examples
Q. 3(6 – 4) =
A. First, distribute the 3 over the 6 – 4: 3(6 – 4) = 3 × 6 – 3 × 4 = 18 – 12 = 6. Another (simpler) way to get the correct answer is just to subtract the 4 from the 6 and then multiply: 3(2) = 6. However, when you can’t or don’t want to combine what’s in the grouping symbols, you use the distributive property.
Q.
A.
Practice Questions
1. 4(7 + y) =
2. –3(x – 11) =
3.
4.
5.
6.
Practice Answers
1.
2.
3.
4. –5.
5.
6.
The associative property has to do with how the numbers are grouped when you perform operations on more than two numbers. Think about what the word associate means. When you associate with someone, you’re close to the person, or you’re in the same group with the person. Say that Anika, Becky, and Cora associate. Whether Anika drives over to pick up Becky and the two of them go to Cora’s and pick her up, or Cora is at Becky’s house and Anika picks up both of them at the same time, the same result occurs – the three ladies are all in the car at the end.
Tip: The associative property means that even if the grouping of the operation changes, the result remains the same. (If you need a reminder about grouping, refer to “Getting a Grip on Grouping Symbols,” earlier in this chapter.) Addition and multiplication are associative. Subtraction and division are not associative operations. So,
You can always find a few cases where the associative property works even though it isn’t supposed to. For example, in the subtraction problem 5 – (4 – 0) = (5 – 4) – 0, the property seems to work. Also, in the division problem 6 ÷ (3 ÷ 1) = (6 ÷ 3) ÷ 1, it seems to work. I just picked numbers very carefully that would make it seem like you could associate with subtraction and division. Although there are exceptions, a rule must work all the time, not just in special cases.
Here’s how the associative property works:
✓ 4 + (5 + 8) = 4 + 13 = 17 and (4 + 5) + 8 = 9 + 8 = 17, so 4 + (5 + 8) = (4 + 5) + 8
✓ 3 – (2 × 5) = 3 × 10 = 30 and (3 × 2) × 5 = 6 × 5 = 30, so 3 × (2 × 5) = (3 × 2) × 5
✓
✓ 3.2 × (5 × 4.8) = 3.2 × 24 = 76.8 and (3.2 × 5) × 4.8 = 16 × 4.8 = 76.8
Remember: This rule is special to addition and multiplication. It doesn’t work for subtraction or division. You’re probably wondering why even use this rule? Because it can sometimes make the computation easier.
✓ Instead of doing 5 + (–5 + 17), change it to [5 + (–5)] + 17 = 0 + 17 = 17.
✓ Instead of