 Succeed with maths – Part 1

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# 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 this activity 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

Timing: Allow approximately 10 minutes

Copy the table below onto a piece of paper and use the exponent button on your calculator to find the value of each of the following. Before you start, think about what pattern you might see and make a note of it.

### Table 4 Exponents

 (–1)2 _ (–2)2 _ (–1)3 _ (–2)3 _ (–1)4 _ (–2)4 _ (–1)5 _ (–2)5 _ (–1)6 _ (–2)6 _

#### Table 4 Exponents (completed)

 (–1)2 1 (–2)2 4 (–1)3 –1 (–2)3 –8 (–1)4 1 (–2)4 16 (–1)5 –1 (–2)5 –32 (–1)6 1 (–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 by enabling rules to be written and mathematical theories to be developed that can be used with any appropriate situation.

The next section is one final recap of all the rules that you have learned involving negative numbers.