2 Calculator exploration: exponents with negative numbers
From your previous study you know that when a number is raised to a certain whole number, exponent or power, that is an instruction to multiply the number by itself as many times as the value of the power. So, 33 = 3 × 3 × 3. In Activity 6 you’re going to explore what happens when the number to be raised to a certain power is negative.
Activity 6 Taking exponents of negative numbers
Use the exponent button on your calculator to find the value of each of the calculations in Table 3. Before you start, think about what pattern you might see and make a note of it.
| (–1)2 | |
| (–1)3 | |
| (–1)4 | |
| (–1)5 | |
| (–1)6 | |
| (–2)2 | |
| (–2)3 | |
| (–2)4 | |
| (–2)5 | |
| (–2)6 |
Answer
| (–1)2 | 1 |
| (–1)3 | –1 |
| (–1)4 | 1 |
| (–1)5 | –1 |
| (–1)6 | 1 |
| (–2)2 | 4 |
| (–2)3 | –8 |
| (–2)4 | 16 |
| (–2)5 | –32 |
| (–2)6 | 64 |
The following patterns should be quite clear in the table:
- raising a negative number to an even exponent gives a positive number
- raising a negative number to an odd exponent gives a negative number.
Looking at what is happening, it should be clear that this was the answer to expect:
- (–2)4 = (–2) × (–2) × (–2) × (–2) = 4 × 4 = 16 (since negative × negative = positive)
- (–2)3 = (–2) × (–2) × (–2) = 4 × (–2) = –8 (since negative × negative = positive, and positive × negative = negative).
There are lots of patterns like these in mathematics. It’s worth watching out for them, as they often give shortcuts to an answer.
You can also see another important feature of maths here – that things link together. The rules for exponents come from the rules about multiplying together negative and/or positive numbers.
The next section is one final recap of all the rules that you have learned involving negative numbers.