Multiplying and dividing by powers of 10
🎯 In this topic you will
- Multiply and divide whole numbers by powers of 10
- Multiply and divide decimals by powers of 10
- Multiply numbers by 10 raised to any positive or negative power
🧠 Key Words
- classify
- index (plural indices)
- powers of 10
- power
Show Definitions
- classify: To arrange or group things into categories based on shared features.
- index (plural indices): A number showing how many times a base is multiplied by itself, also called an exponent.
- powers of 10: Numbers written as 10 raised to an exponent, such as 10² = 100 or 10⁻³ = 0.001.
- power: The result of multiplying a base by itself a certain number of times, indicated by an exponent.
You can write all the numbers $10$, $100$, $1000$, $10000$, … as powers of 10. The power of $10$ is written as an index. This is the number of 10s that you multiply together to get the final number. It is also the same as the number of zeros that follow the digit 1.
Look at this pattern of numbers:
$10 = 10^1$ 10 is ten to the power of 1 or simply 10.
$100 = 10 \times 10 = 10^2$ 100 is ten to the power of 2 or ‘10 squared’.
$1000 = 10 \times 10 \times 10 = 10^3$ 1000 is ten to the power of 3 or ‘10 cubed’.
$10000 = 10 \times 10 \times 10 \times 10 = 10^4$ 10000 is ten to the power of 4.
🔎 Reasoning Tip
Pattern growth: As this pattern continues, the numbers get larger and larger.
When you multiply a number by a power of $10$, you move the digits in the number to the left in the place value table.
When you divide a number by a power of $10$, you move the digits in the number to the right in the place value table.
❓ EXERCISES
1. Write the following in:
i numbers ii words
a. $10^{3}$
b. $10^{5}$
c. $10^{7}$
d. $10^{1}$
👀 Show answer
i numbers
a. $1\,000$ b. $100\,000$ c. $10\,000\,000$ d. $10$
ii words
a. one thousand b. one hundred thousand c. ten million d. ten
2. Write each number as a power of $10$.
a. $100$
b. $100\,000\,000$
c. $10\,000$
d. $10\,000\,000\,000$
👀 Show answer
a. $10^{2}$ b. $10^{8}$ c. $10^{4}$ d. $10^{10}$
3. Copy and complete the working for each of these.
a. $3 \times 10^{4} = 3 \times 10\,000 = \square$
b. $5 \times 10^{6} = 5 \times 1\,000\,000 = \square$
c. $45 \times 10^{5} = 45 \times 100\,000 = \square$
d. $291 \times 10^{3} = 291 \times 1000 = \square$
👀 Show answer
a. $30\,000$ b. $5\,000\,000$ c. $4\,500\,000$ d. $291\,000$
4. Marcus says: ‘When I multiply a whole number by $10^{4}$, I just have to add four zeros to the end of the number. When I multiply a whole number by $10^{5}$, I just have to add five zeros to the end of the number. This works for any positive whole number power.’
Is Marcus correct? Discuss in pairs or groups.
👀 Show answer
Yes, for whole numbers. Multiplying a whole number by $10^{n}$ shifts its digits $n$ places to the left, which is the same as appending $n$ zeros (e.g., $37 \times 10^{4} = 370\,000$). This rule is specific to whole numbers in base $10$; numbers with decimals require shifting the decimal point instead of just adding zeros.
5. Work out:
a. $23 \times 10^{2}$
b. $768 \times 10^{4}$
c. $9 \times 10^{6}$
👀 Show answer
a. $2\,300$ b. $7\,680\,000$ c. $9\,000\,000$
6. Copy and complete the working for each of these.
a. $4.2 \times 10^{2} = 4.2 \times 100 = \square$
b. $6.5 \times 10^{4} = 6.5 \times 10\,000 = \square$
c. $12.7 \times 10^{3} = 12.7 \times 1000 = \square$
d. $2.87 \times 10^{6} = 2.87 \times 1\,000\,000 = \square$
👀 Show answer
a. $420$ b. $65\,000$ c. $12\,700$ d. $2\,870\,000$
🧠 Think like a Mathematician
Question: Sofia and Maya use different methods to work out $4.56 \times 10^5$. Which method is clearer and why?
Equipment: Pencil, paper, calculator
Method:
Sofia’s method:
There are five zeros in $10^5$, so move the decimal point five places to the right.
There are three empty spaces, so fill them with zeros.
So, $4.56 \times 10^5 = 456000$
Maya’s method:
The number 4.56 has two decimal places.
The number $10^5$ has five zeros.
$5 - 2 = 3$, so I must add three zeros to the end of the number and remove the decimal point.
So, $4.56 \times 10^5 = 456000$
- a) Do you understand how both methods work?
- b) Critique the two methods. Explain the advantages and disadvantages of each method.
- c) Which method do you prefer?
- d) Why doesn’t Marcus’s method work for this type of question? Discuss in pairs or groups.
Follow-up Questions:
👀 show answer
- a: Yes. Sofia moves the decimal point, while Maya counts decimal places and adds zeros.
- b: Sofia’s method is visual and works well for understanding place value. Maya’s method is efficient but requires knowing decimal places quickly.
- c: Preference depends on the learner — visual thinkers may like Sofia’s, while those confident with place values may prefer Maya’s.
- d: Marcus’s method may not work if it relies on a step that assumes a fixed decimal length or doesn’t account for the correct movement of the decimal point.
❓ EXERCISES
8. Work out:
a. $4.7 \times 10^{4}$
b. $91.5 \times 10^{3}$
c. $0.33 \times 10^{7}$
👀 Show answer
a. $47\,000$ b. $91\,500$ c. $3\,300\,000$
9. Copy these calculations and fill in the missing numbers.
a. $1.5 \times 10^{3} = \square$
b. $32.1 \times \square = 3\,210$
c. $\square \times 10^{5} = 612\,000$
d. $124.63 \times 10 \times \square = 12\,463\,000$
👀 Show answer
a. $1\,500$ b. $100$ c. $6.12$ d. $1\,000$
10. Copy and complete the working for each of the following.
a. $8000 + 10^{3} = 8000 + 1000 = \square$
b. $80\,500\,000 + 10^{5} = 80\,500\,000 + 100\,000 = \square$
👀 Show answer
a. $9\,000$ b. $80\,600\,000$
11. Marcus says: ‘When I divide a whole number by $10^{4}$, I just have to cross four zeros off the end of the number. When I divide a whole number by $10^{5}$, I just have to cross five zeros off the end of the number. This works for any positive whole number power.’
Is Marcus correct? Discuss in pairs or groups.
👀 Show answer
Yes, for whole numbers. Dividing a whole number by $10^{n}$ shifts its digits $n$ places to the right, which is the same as removing $n$ zeros from the end (e.g., $37\,000 \div 10^{3} = 37$). This rule is specific to whole numbers; numbers with decimals require moving the decimal point instead of just removing zeros.
12. Work out:
a. $80\,000 \div 10^{4}$
b. $510\,000 \div 10^{3}$
c. $8\,460\,000\,000 \div 10^{5}$
👀 Show answer
a. $8$ b. $510$ c. $84\,600$
🧠 Think like a Mathematician
Question: Sofia and Maya use different methods to work out $2\,350\,000 \div 10^5$. Which method do you think is more effective?
Equipment: Pencil, paper, calculator
Method:
Sofia’s method:
There are five zeros in $10^5$, so move the decimal point five places to the left.
The decimal point stops between the 3 and the 5.
So, $2\,350\,000 \div 10^5 = 23.5$
Maya’s method:
The number 2,350,000 has four zeros.
The number $10^5$ has five zeros.
$5 - 4 = 1$, so there must be one digit (non-zero) after the decimal point.
So, $2\,350\,000 \div 10^5 = 23.5$
- a) Do you understand how both methods work?
- b) Critique the two methods by explaining the advantages and disadvantages of each.
- c) Which method do you prefer?
- d) Can you think of a better method to use for this type of question?
Follow-up Questions:
👀 show answer
- a: Yes. Sofia moves the decimal point left; Maya calculates the difference between the zeros in the number and in $10^n$ to determine decimal placement.
- b: Sofia’s method is more visual and easier for beginners, but can be slower for large numbers. Maya’s method is faster if you quickly recognise place values but requires confidence with counting zeros.
- c: Preference depends on familiarity — visual learners may prefer Sofia’s; mental calculators may prefer Maya’s.
- d: A calculator or scientific notation method might be quicker for very large or very small numbers.
❓ EXERCISES
14. Work out:
a. $230 \div 10^{1}$ b. $230 \div 10^{2}$ c. $230 \div 10^{3}$ d. $230 \div 10^{4}$
e. $65 \div 10^{1}$ f. $65 \div 10^{2}$ g. $65 \div 10^{3}$ h. $65 \div 10^{4}$
i. $9 \div 10^{1}$ j. $9 \div 10^{2}$ k. $9 \div 10^{3}$ l. $9 \div 10^{4}$
👀 Show answer
a. $23$ b. $2.3$ c. $0.23$ d. $0.023$
e. $6.5$ f. $0.65$ g. $0.065$ h. $0.0065$
i. $0.9$ j. $0.09$ k. $0.009$ l. $0.0009$
15. Choose the correct answer, A, B or C, for each of the following.
a. $6700000 \div 10^{5}$ A $670$ B $6.7$ C $0.67$
b. $9520 \div 10^{4}$ A $0.952$ B $9.52$ C $0.0952$
c. $18500000 \div 10^{6}$ A $185$ B $1.85$ C $18.5$
👀 Show answer
a. A ($670$) b. B ($9.52$) c. A ($185$)
16. These formulae show how to convert between some metric units of length.
Use the formulae to work out the missing numbers in these conversions:
a. $\square \ \text{mm} = 8 \ \text{cm}$
b. $\square \ \text{mm} = 15 \ \text{cm}$
c. $\square \ \text{mm} = 7 \ \text{m}$
d. $\square \ \text{mm} = 3.4 \ \text{m}$
e. $\square \ \text{mm} = 9 \ \text{km}$
f. $\square \ \text{mm} = 0.6 \ \text{km}$
g. $\square \ \text{mm} = 12.4 \ \text{cm}$
h. $\square \ \text{mm} = 32.25 \ \text{km}$
👀 Show answer
a. $80 \ \text{mm}$ b. $150 \ \text{mm}$ c. $7000 \ \text{mm}$ d. $3400 \ \text{mm}$
e. $9000000 \ \text{mm}$ f. $600000 \ \text{mm}$ g. $124 \ \text{mm}$ h. $32250000 \ \text{mm}$
17.
a. Convert $8000000 \ \text{mm}$ to km.
🔎 Reasoning Tip
Unit conversions: Start by converting millimetres (mm) to centimetres (cm), then centimetres to metres (m), and finally metres to kilometres (km).
b. Write a formula that will convert a length in mm to a length in km.
c. Use your formula from part b to work out the missing numbers in these conversions:
i. $\square \ \text{km} = 90000000 \ \text{mm}$
ii. $\square \ \text{km} = 15600000 \ \text{mm}$
iii. $\square \ \text{km} = 770000 \ \text{mm}$
👀 Show answer
a. $8 \ \text{km}$
b. $\text{km} = \frac{\text{mm}}{10^{6}}$
c. i. $90 \ \text{km}$ ii. $15.6 \ \text{km}$ iii. $0.77 \ \text{km}$
18. a. Classify these number cards into groups of the same value. There should be one card left over.
b. Write two new cards that have the same value as the card that is left over.
👀 Show answer
a. Groupings vary, but one card (likely $0.00078 \times 10^{6}$) will be left over.
b. Example: $7.8 \times 10^{2}$ and $780 \times 10^{0}$
🍬 Learning Bridge
Now that you can multiply and divide whole numbers and decimals by positive powers of 10, you're ready to extend this understanding to negative powers. This will show how numbers smaller than 1 can also be expressed as powers of 10, linking decimals, fractions, and index notation into one clear pattern.
Look at this section of the decimal place-value table.
| ... | thousands | hundreds | tens | units | • | tenths | hundredths | thousandths | ... |
| ... | $1000$ | $100$ | $10$ | $1$ | • | $\frac{1}{10}$ | $\frac{1}{100}$ | $\frac{1}{1000}$ | ... |
You can write the numbers $10$, $100$, $1000$, … as positive powers of 10.
You can write the numbers $\frac{1}{10}$, $\frac{1}{100}$, $\frac{1}{1000}$, … as negative powers of 10.
Look at this pattern of numbers, written as powers of 10. Is there a link between the powers and the values?
..., $1000 = 10^3$, $100 = 10^2$, $10 = 10^1$, $1 = 10^0$, $\frac{1}{10} = 10^{-1}$, $\frac{1}{100} = 10^{-2}$, $\frac{1}{1000} = 10^{-3}$, ...
🔎 Reasoning Tip
Decimals as fractions and powers of ten: You can write the decimal \(0.1\) as \(\frac{1}{10}\) or \(10^{-1}\). You can write the decimal \(0.01\) as \(\frac{1}{100}\) or \(10^{-2}\).
This pattern continues as the numbers get bigger and smaller.
For example, $10000 = 10^4$ and $\frac{1}{10000} = 10^{-4}$, $100000 = 10^5$ and $\frac{1}{100000} = 10^{-5}$.
It is important to remember these two key points:
1Multiplying a number by $\frac{1}{10}, \frac{1}{100}, \frac{1}{1000}, \frac{1}{10000}$ … is the same as dividing the same number by $10, 100, 1000, \dots$
2Dividing a number by $\frac{1}{10}, \frac{1}{100}, \frac{1}{1000}, \frac{1}{10000}$ … is the same as multiplying the same number by $10, 100, 1000, \dots$
❓ EXERCISES
19. Match each decimal with the correct fraction and power of $10$. The first one has been done for you: a, D and ii.
| Decimal | Fraction | Power of $10$ |
|---|---|---|
| $0.1$ | $\frac{1}{10}$ | $10^{-1}$ |
| $0.001$ | $\frac{1}{1000}$ | $10^{-3}$ |
| $0.00001$ | $\frac{1}{100000}$ | $10^{-5}$ |
| $0.01$ | $\frac{1}{100}$ | $10^{-2}$ |
| $0.0001$ | $\frac{1}{10000}$ | $10^{-4}$ |
👀 Show answer
a – D – ii ($0.1 = \frac{1}{10} = 10^{-1}$)
b – A – v ($0.001 = \frac{1}{1000} = 10^{-3}$)
c – E – iv ($0.00001 = \frac{1}{100000} = 10^{-5}$)
d – C – i ($0.01 = \frac{1}{100} = 10^{-2}$)
e – B – iii ($0.0001 = \frac{1}{10000} = 10^{-4}$)
20. Copy and complete
a. $3.2 \times 10^{3} = 3.2 \times 1000 = \ldots$
b. $3.2 \times 10^{2} = 3.2 \times 100 = \ldots$
c. $3.2 \times 10^{1} = 3.2 \times \ldots = \ldots$
d. $3.2 \times 10^{0} = 3.2 \times 1 = \ldots$
e. $3.2 \times 10^{-1} = 3.2 \div 10 = \ldots$
f. $3.2 \times 10^{-2} = 3.2 \div 100 = \ldots$
g. $3.2 \times 10^{-3} = 3.2 \div \ldots = \ldots$
h. $3.2 \times 10^{-4} = 3.2 \div \ldots = \ldots$
👀 Show answer
a. $3.2 \times 10^{3} = 3.2 \times 1000 = 3200$
b. $3.2 \times 10^{2} = 3.2 \times 100 = 320$
c. $3.2 \times 10^{1} = 3.2 \times 10 = 32$
d. $3.2 \times 10^{0} = 3.2 \times 1 = 3.2$
e. $3.2 \times 10^{-1} = 3.2 \div 10 = 0.32$
f. $3.2 \times 10^{-2} = 3.2 \div 100 = 0.032$
g. $3.2 \times 10^{-3} = 3.2 \div 1000 = 0.0032$
h. $3.2 \times 10^{-4} = 3.2 \div 10000 = 0.00032$
21. Look at your answers to Question $2$. Compare all your answers with the number you started with, $3.2$. Arun makes this conjecture.
When you multiply $3.2$ by $10$ to a negative power, the answer is always smaller than $3.2$.
a. Is Arun correct? Use specialising to explain your answer.
b. Copy and complete these generalising statements:
i. When you multiply a number by $10$ to a negative power, the answer is always ……… than the number you started with.
ii. When you multiply a number by $10$ to the power zero, the answer is always ……… as the number you started with.
iii. When you multiply a number by $10$ to a positive power, the answer is always ……… than the number you started with.
👀 Show answer
a. Yes. For any positive integer $n$, $10^{-n}=\dfrac{1}{10^{n}} \lt 1$, so $3.2 \times 10^{-n}$ is less than $3.2$ (e.g., $0.32$, $0.032$, $0.0032$ in Question $2$).
b. i. smaller ii. the same iii. greater
22. Work out:
a. $13 \times 10^{2}$ b. $7.8 \times 10^{3}$ c. $24 \times 10^{1}$
d. $8.55 \times 10^{4}$ e. $6.5 \times 10^{1}$ f. $0.08 \times 10^{5}$
g. $17 \times 10^{0}$ h. $8 \times 10^{-1}$ i. $8.5 \times 10^{-2}$
j. $4500 \times 10^{-4}$ k. $32 \times 10^{-3}$ l. $125 \times 10^{-2}$
👀 Show answer
a. $1300$ b. $7800$ c. $240$
d. $85500$ e. $65$ f. $8000$
g. $17$ h. $0.8$ i. $0.085$
j. $0.45$ k. $0.032$ l. $1.25$
23. Copy and complete:
a. $320 \div 10^{3} = 320 \div 1000 = \ldots$
b. $320 \div 10^{2} = 320 \div 100 = \ldots$
c. $320 \div 10^{1} = 320 \div \ldots = \ldots$
d. $320 \div 10^{0} = 320 \div 1 = \ldots$
👀 Show answer
a. $0.32$ b. $3.2$ c. $32$ d. $320$
24. Work out:
a. $27 \div 10^{1}$ b. $450 \div 10^{3}$ c. $36 \div 10^{2}$
d. $170 \div 10^{4}$ e. $0.8 \div 10^{1}$ f. $2480 \div 10^{5}$
g. $9 \div 10^{0}$ h. $0.25 \div 10^{2}$
👀 Show answer
a. $2.7$ b. $0.45$ c. $0.36$
d. $0.017$ e. $0.08$ f. $0.0248$
g. $9$ h. $0.0025$
🧠 Think like a Mathematician
Question: Two students use different methods to evaluate $2.6 \div 10^{-2}$. What do their methods show, and which approach would you choose for similar problems?
Equipment: Pencil, paper, calculator (optional)
Method (worked examples):
Cesar’s method
$10^{-2}=\dfrac{1}{10^2}=\dfrac{1}{100}$
$2.6 \div \dfrac{1}{100}=2.6 \times 100=260$
Domonique’s method
$2.6=\dfrac{26}{10}=26\times10^{-1}$
$(26\times10^{-1}) \div 10^{-2}=26\times\dfrac{10^{-1}}{10^{-2}}=26\times10^{-1-(-2)}=26\times10^{1}=260$
- a) Explain in your own words how each method works.
- b) Use both methods to work out:
- $6.8 \div 10^{-3}$
- $0.07 \div 10^{-4}$
- c) Which method do you prefer? Give reasons.
- d) Suggest an even quicker general method for dividing a decimal by $10^{-k}$.
- e) Write a short reflection comparing the methods and your choice in part c.
Follow-up prompts:
👀 show answer
- b.i:$6.8 \div 10^{-3}=6.8 \times 10^{3}=6800$
- b.ii:$0.07 \div 10^{-4}=0.07 \times 10^{4}=700$
- c (sample): Preference: Domonique’s method — it uses the index law directly and scales to any power of ten without rewriting as a fraction.
- d: Use the rule $x \div 10^{-k}=x \times 10^{k}$. In words: dividing by a negative power of 10 is the same as multiplying by the corresponding positive power; move the decimal point right by $k$ places.
- Connection: Both methods rely on $10^{-k}=\dfrac{1}{10^{k}}$ and the index law $10^{a}\div10^{b}=10^{a-b}$.
❓ EXERCISES
26. Copy and complete
a. $3.2 \div 10^{3} = 3.2 \div 1000 = \ldots$
b. $3.2 \div 10^{2} = 3.2 \div 100 = \ldots$
c. $3.2 \div 10^{1} = 3.2 \div \ldots = \ldots$
d. $3.2 \div 10^{0} = 3.2 \div 1 = \ldots$
e. $3.2 \div 10^{-1} = 3.2 \times 10 = \ldots$
f. $3.2 \div 10^{-2} = 3.2 \times 100 = \ldots$
g. $3.2 \div 10^{-3} = 3.2 \times \ldots = \ldots$
h. $3.2 \div 10^{-4} = 3.2 \times \ldots = \ldots$
👀 Show answer
a. $0.0032$ b. $0.032$ c. $3.2 \div 10 = 0.32$ d. $3.2$
e. $32$ f. $320$ g. $3.2 \times 1000 = 3200$ h. $3.2 \times 10000 = 32000$
27. Look at your answers to Question $8$. Compare all your answers with the number you started with, $3.2$. Zara makes this conjecture.
When you divide $3.2$ by $10$ to a negative power, the answer is always greater than $3.2$.
a. Is Zara correct? Use specialising to explain your answer.
b. Copy and complete these generalising statements:
i. When you divide a number by $10$ to a negative power, the answer is always ……… than the number you started with.
ii. When you divide a number by $10$ to the power zero, the answer is always ……… as the number you started with.
iii. When you divide a number by $10$ to a positive power, the answer is always ……… than the number you started with.
👀 Show answer
a. Yes. Since $10^{-n}=\dfrac{1}{10^{n}}\lt 1$, dividing by $10^{-n}$ is the same as multiplying by $10^{n}$, which makes the result larger (e.g., from Question $8$: $32$, $320$, $3200$, $32000$).
b. i. greater ii. the same iii. smaller
28. Work out:
a. $0.25 \div 10^{-1}$ b. $4.76 \div 10^{-4}$
c. $0.07 \div 10^{-3}$ d. $0.085 \div 10^{-2}$
👀 Show answer
a. $2.5$ b. $47600$ c. $70$ d. $8.5$
29. Copy this table, which contains a secret coded message.
| 3 | 3.3 | 0.3 | 3.3 | 300 | 33 | 6 | 6 | 0.6 | 0.3 | 0.33 | 3.3 | 0.3 | 33 | 300 | 0.06 | 33 | 60 | 33 | 0.03 | 600 | 33 | 300 |
Work out the answers to the calculations in the code box below. Find the answer in your secret code table. Write the letter from the code box above the number in your table.
| $0.6 \times 10^{2} = S$ | $60 \times 10^{-1} = L$ | $0.06 \div 10^{-1} = A$ | $600 \div 10^{0} = R$ | $60 \times 10^{-3} = H$ | $33 \div 10^{1} = O$ | $0.33 \times 10^{0} = Y$ | $300 \times 10^{-3} = N$ | $300 \div 10^{4} = C$ | $3300 \times 10^{-2} = E$ | $0.3 \div 10^{-3} = T$ | $300 \div 10^{2} = D$ |
👀 Show answer
Secret coded message: "HAPPY LEARNING!"
This is found by solving each number in the orange table using the green code box and writing the corresponding letter above each number.
🧠 Think like a Mathematician
Question: How does multiplying or dividing by powers of 10 affect the size of a number?
Equipment: Pencil, paper, calculator (optional)
Method:
- a) Work out:
- $4 \times 10^2$
- $4 \times 10^1$
- $4 \times 10^0$
- $4 \times 10^{-1}$
- $4 \times 10^{-2}$
- $4 \times 10^{-3}$
- b) Use specialising to answer: When you multiply a number by $10^{-4}$, is the answer larger or smaller than when you multiply it by $10^{-3}$?
- c) Complete the sentence: When you multiply a number by $10^x$, the smaller the power, the ________ the answer.
- d) Work out:
- $12 \div 10^2$
- $12 \div 10^1$
- $12 \div 10^0$
- $12 \div 10^{-1}$
- $12 \div 10^{-2}$
- $12 \div 10^{-3}$
- e) Use specialising to answer: When you divide a number by $10^{-4}$, is the answer larger or smaller than when you divide it by $10^{-3}$?
- f) Complete the sentence: When you divide a number by $10^x$, the smaller the power, the ________ the answer.
- g) Discuss your answers with others.
👀 show answer
- a: i) $400$ ii) $40$ iii) $4$ iv) $0.4$ v) $0.04$ vi) $0.004$
- b: Multiplying by $10^{-4}$ gives a smaller result than multiplying by $10^{-3}$.
- c: smaller.
- d: i) $0.12$ ii) $1.2$ iii) $12$ iv) $120$ v) $1200$ vi) $12000$
- e: Dividing by $10^{-4}$ gives a larger result than dividing by $10^{-3}$.
- f: larger.
❓ EXERCISES
31. Work out the missing power in each question in these spider diagrams.
In each part, all the questions in the outer shapes should give the answer in the centre shape.
a.
b.
👀 Show answer
a (centre = $8$):
$0.8 \times 10^{\color{black}{1}} = 8$, $8 \div 10^{\color{black}{0}} = 8$, $80 \times 10^{\color{black}{-1}} = 8$,
$0.08 \div 10^{\color{black}{-2}} = 8$, $0.008 \times 10^{\color{black}{3}} = 8$, $800 \div 10^{\color{black}{2}} = 8$
b (centre = $0.32$):
$32 \div 10^{\color{black}{2}} = 0.32$, $0.32 \times 10^{\color{black}{0}} = 0.32$, $3.2 \div 10^{\color{black}{1}} = 0.32$,
$320 \div 10^{\color{black}{3}} = 0.32$, $32 \times 10^{\color{black}{-2}} = 0.32$, $3.2 \times 10^{\color{black}{-1}} = 0.32$
⚠️ Be careful!
Dividing by $10^{-n}$ makes the number bigger, not smaller. For example, $2.5 \div 10^{-3} = 2.5 \times 10^3 = 2500$. Remember: a negative index means a reciprocal, so dividing by it reverses the effect.