# 2.6.1 Try some yourself

## Activity 28

Find each of the following by hand, giving your answers both as a power of ten and as a decimal number. You will use these answers as a check on your calculator work in the next question.

(a) 10

^{−2}(b) 10

^{2}× 10^{3}(c) 10

^{7}÷ 10^{4}(d) 10

^{4}÷ 10^{7}(e) 2

^{−2}

### Answer

(a) 10

^{−2}= 0.01(b) 10

^{2}× 10^{3}= 10^{2 + 3}= 10^{5}= 100 000(c) 10

^{7}÷ 10^{4}= 10^{7 − 4}= 10^{3}= 1000(d) 10

^{4}÷ 10^{7}= 10^{4 − 7}= 10^{–3}= = 0.001(e) 2

^{−2}= = (=0.25)

## Activity 29

Evaluate the following, using your calculator and your answers from Question 1 as estimates.

(a) 10.3

^{−2}(to four significant figures)(b) 10.1

^{2}× 10.11^{3}(to four significant figures)(c) 10.112

^{7}÷ 10.21^{4}(to one decimal place)(d) 10.12

^{4}÷ 10.351^{7}(to three significant figures)(e) 2.2

^{−2}

### Answer

Using Question 1 as a rough check, the calculator gives:

(a) 10.3

^{−2}= 0.009 426 (to 4 s.f.)(b) 10.1

^{2}× 10.11^{3}= 105 400 (to 4 s.f.)(c) 10.112

^{7}÷ 10.21^{4}= 994.8 (to 1 d.p.)(d) 10.12

^{4}÷ 10.351^{7}= 0.000824 (to 3 s.f.)(e) 2.2

^{−2}0.2066

It is a good idea to get in the habit of estimating your answers, even if not specifically asked to do so.

## Activity 30

Find each of the following by hand.

(a) 1024

^{0}(b) 1024

^{1}(c) 5

^{−1}(d) 10

^{−4}

### Answer

(a) 1024

^{0}= 1(b) 1024

^{1}= 1024(c)

(d)

## Activity 31

How many zeros are there after the decimal point in the number 10^{−6}? How many zeros are there after the decimal point for any given negative power of 10?

### Answer

10^{−6} = 0.000 001; there are five zeros after the decimal point. For any given negative power of 10, say 10^{−(NUMBER)} there is one fewer zero than NUMBER. This works for 10^{−1}, too, since 10^{−1} = 0.1, which has no zeros after the decimal point, i.e. one fewer than 1.

## Activity 32

Try to answer the following questions

(a) Explain why a negative power of a number is one divided by the corresponding positive power of that number. (

*Hint*: remember that a number to the power zero is 1.)(b) Use the power button on your calculator to find the following.

(i)

(ii)

### Answer

(a) Take an example, which makes the explanation easier than describing it in the abstract. Consider 10

^{−6}. Now^{−}6 is just the same as 0 − 6, so 10^{−6}= 10^{0−6}. Using the rule for dividing powers, 10^{0−6}= 10^{0}÷ 10^{6}. But 10^{0}= 1. So 10^{−6}= 1 ÷ 10^{6}. This argument would apply equally well to any number, since any number to the power 0 is 1. (Alternatively you may have started from the other end, for example, by showing that = = 10^{0−6}= 10^{−6}.)(b)