Year 5: Square numbers and cube numbers
Part of the Year 5 prime numbers, factors, squares and cubes chapter. Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Square numbers and cube numbers
When you multiply a number by itself, you get a square number. 3 × 3 = 9, so 9 is a square number. If you arrange 9 dots or tiles in 3 rows of 3, they make a perfect square. We write 3 × 3 as 3² and say “3 squared”. The little 2 means “multiply the number by itself”. It does not mean “times 2”: 3² = 9, not 6.
You should know the squares up to 12 × 12 straight away. They are in the table. Notice the pattern: you add the next odd number each time (1 + 3 = 4, 4 + 5 = 9, 9 + 7 = 16…).
| Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Squared | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 |
Squares beyond 12. Some bigger squares are very useful: 15² = 225, 20² = 400, 25² = 625 and 30² = 900. Here are three tricks.
- Tens. 20² = 2 × 2 × 10 × 10 = 4 × 100 = 400. So 30² = 900, 40² = 1,600 and 50² = 2,500.
- Partition. 15² = 15 × 10 + 15 × 5 = 150 + 75 = 225.
- Ending in 5. Take the tens digit, multiply it by the next number, and write 25 on the end: 25² → 2 × 3 = 6, so 625. 35² → 3 × 4 = 12, so 1,225.
Cube numbers. When you multiply a number by itself and then by itself again, you get a cube number. 2 × 2 × 2 = 8. We write it 2³ and say “2 cubed”. A cube number is the number of small cubes in a big cube. A 2 by 2 by 2 cube is made from 8 small cubes. A 3 by 3 by 3 cube has 27.
| Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| Cubed | 1 | 8 | 27 | 64 | 125 | 216 | 343 | 512 | 729 | 1,000 |
A few numbers are both square and cube: 1, 64 (8² and 4³) and 729 (27² and 9³).
Key ideas
- A square number is a number multiplied by itself: 6² = 6 × 6 = 36. The little 2 does not mean “times 2”.
- Squares up to 12²: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. Also know 15² = 225, 20² = 400, 25² = 625, 30² = 900.
- A cube number is a number multiplied by itself twice: 4³ = 4 × 4 × 4 = 64. The little 3 does not mean “times 3”.
- Cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1,000.
- To test if a number is square (or cube), list the squares (or cubes) and see if it is there.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. 7² means 7 multiplied by itself: 7 × 7.
- Say what the little 2 means. The little 2 means “multiply the number by itself”. It does not mean “times 2”.7² = 7 × 7
- Use a times table fact. You know this from the times tables.7 × 7 = 49
- Picture it. 7 rows of 7 dots make a square.7 rows × 7 columns = 49 dots
Check. The square numbers go up by the odd numbers: 36 + 13 = 49. ✓
Watch out. 7² is not 7 × 2 = 14. Multiply 7 by 7.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. 15² means 15 multiplied by itself: 15 × 15.
- Say what the little 2 means. The little 2 means “multiply the number by itself”. It does not mean “times 2”.15² = 15 × 15
- Split the second 15 into 10 and 5. Multiply 15 by each part. This is called partitioning.15 × 15 = 15 × 10 + 15 × 5
15 × 10 = 150
15 × 5 = 75 - Add the two answers. 150 + 75 = 225
Check. Pattern for numbers ending in 5: 1 × 2 = 2, then write 25 on the end: 225. ✓
Watch out. 15² is not 15 × 2 = 30. Multiply 15 by 15.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. 20² means 20 multiplied by itself: 20 × 20.
- Say what the little 2 means. The little 2 means “multiply the number by itself”. It does not mean “times 2”.20² = 20 × 20
- Split each number into a single digit and a 10. 20 is 2 × 10. Multiplying can be done in any order.20 × 20 = (2 × 10) × (2 × 10) = (2 × 2) × (10 × 10)
- Multiply. 2 × 2 = 4 and 10 × 10 = 100.4 × 100 = 400
Check. 20 × 2 = 40, and 40 × 10 = 400. ✓
Watch out. 20² is not 20 × 2 = 40. Multiply 20 by 20.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. 25² means 25 multiplied by itself: 25 × 25.
- Say what the little 2 means. The little 2 means “multiply the number by itself”. It does not mean “times 2”.25² = 25 × 25
- Split the second 25 into 20 and 5. Multiply 25 by each part. This is called partitioning.25 × 25 = 25 × 20 + 25 × 5
25 × 20 = 500
25 × 5 = 125 - Add the two answers. 500 + 125 = 625
Check. Pattern for numbers ending in 5: 2 × 3 = 6, then write 25 on the end: 625. ✓
Watch out. 25² is not 25 × 2 = 50. Multiply 25 by 25.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. Write the square numbers in order, and see if 50 is one of them.
- List the square numbers. Keep going until you reach or pass 50.1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25, 6² = 36, 7² = 49, 8² = 64
- Look for the number. 50 is between 49 and 64. It is not in the list.
- Answer the question. 50 is not a square number: No
Check. No whole number multiplied by itself makes 50. ✓
Watch out. Do not guess from the last digit. Check the list of square numbers.
Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.
Method. 4³ means 4 × 4 × 4. Multiply the first two, then multiply the answer by 4 again.
- Say what the little 3 means. The little 3 means “multiply three of the number together”. It does not mean “times 3”.4³ = 4 × 4 × 4
- Multiply the first two numbers. Work from left to right.4 × 4 = 16
- Multiply that answer by 4 again. 16 × 4 = 64
Check. Picture a cube: 4 layers, each with 16 small cubes. 4 × 16 = 64. ✓
Watch out. 4³ is not 4 × 3 = 12. It is 4 × 4 × 4.
Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.
Method. 6³ means 6 × 6 × 6. Multiply the first two, then multiply the answer by 6 again.
- Say what the little 3 means. The little 3 means “multiply three of the number together”. It does not mean “times 3”.6³ = 6 × 6 × 6
- Multiply the first two numbers. Work from left to right.6 × 6 = 36
- Multiply that answer by 6 again. 36 × 6 = 30 × 6 + 6 × 6 = 180 + 36 = 216
Check. Picture a cube: 6 layers, each with 36 small cubes. 6 × 36 = 216. ✓
Watch out. 6³ is not 6 × 3 = 18. It is 6 × 6 × 6.
Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.
Method. Write the cube numbers in order, and see if 125 is one of them.
- List the cube numbers. Keep going until you reach or pass 125.1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125
- Look for the number. 125 is in the list.
- Answer the question. 125 is a cube number: Yes
Check. 5 × 5 × 5 = 125. ✓
Watch out. Do not guess from the last digit. Check the list of cube numbers.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. Check the square numbers, then check the cube numbers.
- Is it a square number? List squares until you reach or pass the number.1² = 1 2² = 4 3² = 9 4² = 16
5² = 25 6² = 36 7² = 49 8² = 64
9² = 81
yes - Is it a cube number? List cubes until you reach or pass the number.1³ = 1 2³ = 8 3³ = 27 4³ = 64
5³ = 125
yes - Decide. 64 is a square number and a cube number.
Check. 8 × 8 = 64. ✓
Watch out. A number can be both a square and a cube. 64 = 8 × 8 and 64 = 4 × 4 × 4.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. Work out each power first. Then do the addition or subtraction.
- Work out 3². 3² means 3 × 3.3² = 3 × 3 = 9
- Work out 2³. 2³ means 2 × 2 × 2.2³ = 2 × 2 × 2 = 8
- Now add. Use the two answers.9 + 8 = 17
Check. Estimate: 9 is about 10 and 8 is about 10, so the answer should be near 20. ✓
Watch out. Do not multiply the number by the little number: 3² is not 3 × 2.
Watch out
3² is not 3 × 2 and 3³ is not 3 × 3. 3² = 3 × 3 = 9, and 3³ = 3 × 3 × 3 = 27. Also, do not mix up squares and cubes: 4² = 16 but 4³ = 64. And remember that 25² is not 50: it is 25 × 25 = 625.Exercise 3G Square numbers
Non-calculatorKey words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. Each square is made from the one before by adding an L-shape. Or just work out 6 × 6.
- Spot the pattern. The squares are 1 × 1, 2 × 2, 3 × 3, 4 × 4 and 5 × 5.1, 4, 9, 16, 25
- Find the sixth square. The sixth square has 6 rows of 6 dots.6 × 6 = 36
Check. 25 + 11 = 36. ✓
Watch out. The sixth square is 6 × 6, not 5 × 6.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. The extra dots make an L-shape round the old square. Take the 5th square away from the 6th.
- Write both squares. 5th square: 25 dots 6th square: 36 dots
- Subtract. The difference is the number of extra dots.36 − 25 = 11
Check. The L-shape has 6 + 5 = 11 dots (6 along the bottom and 5 up the side). ✓
Watch out. The extra dots are always the next odd number: 3, 5, 7, 9, 11…
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. Write 6 × 6 using the squared notation.
- Write the multiplication. 6 × 6 = 36
- Use the squared notation. The little 2 shows that the number is multiplied by itself.6² = 36
Check. 6 × 6 = 36. ✓
Watch out. 6² means 6 × 6, not 6 × 2.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. 3² means 3 multiplied by itself: 3 × 3.
- Say what the little 2 means. The little 2 means “multiply the number by itself”. It does not mean “times 2”.3² = 3 × 3
- Use a times table fact. You know this from the times tables.3 × 3 = 9
- Picture it. 3 rows of 3 dots make a square.3 rows × 3 columns = 9 dots
Check. The square numbers go up by the odd numbers: 4 + 5 = 9. ✓
Watch out. 3² is not 3 × 2 = 6. Multiply 3 by 3.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. 5² means 5 multiplied by itself: 5 × 5.
- Say what the little 2 means. The little 2 means “multiply the number by itself”. It does not mean “times 2”.5² = 5 × 5
- Use a times table fact. You know this from the times tables.5 × 5 = 25
- Picture it. 5 rows of 5 dots make a square.5 rows × 5 columns = 25 dots
Check. The square numbers go up by the odd numbers: 16 + 9 = 25. ✓
Watch out. 5² is not 5 × 2 = 10. Multiply 5 by 5.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. 8² means 8 multiplied by itself: 8 × 8.
- Say what the little 2 means. The little 2 means “multiply the number by itself”. It does not mean “times 2”.8² = 8 × 8
- Use a times table fact. You know this from the times tables.8 × 8 = 64
- Picture it. 8 rows of 8 dots make a square.8 rows × 8 columns = 64 dots
Check. The square numbers go up by the odd numbers: 49 + 15 = 64. ✓
Watch out. 8² is not 8 × 2 = 16. Multiply 8 by 8.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. 11² means 11 multiplied by itself: 11 × 11.
- Say what the little 2 means. The little 2 means “multiply the number by itself”. It does not mean “times 2”.11² = 11 × 11
- Use a times table fact. You know this from the times tables.11 × 11 = 121
- Picture it. 11 rows of 11 dots make a square.11 rows × 11 columns = 121 dots
Check. The square numbers go up by the odd numbers: 100 + 21 = 121. ✓
Watch out. 11² is not 11 × 2 = 22. Multiply 11 by 11.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. 12² means 12 multiplied by itself: 12 × 12.
- Say what the little 2 means. The little 2 means “multiply the number by itself”. It does not mean “times 2”.12² = 12 × 12
- Use a times table fact. You know this from the times tables.12 × 12 = 144
- Picture it. 12 rows of 12 dots make a square.12 rows × 12 columns = 144 dots
Check. The square numbers go up by the odd numbers: 121 + 23 = 144. ✓
Watch out. 12² is not 12 × 2 = 24. Multiply 12 by 12.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. Try the numbers 1, 2, 3, … in order. Work out each square until you reach 49.
- Write the square numbers in order. Stop when you reach the number you want.1² = 1 2² = 4 3² = 9 4² = 16
5² = 25 6² = 36 7² = 49 - Find 49 in the list. 7² = 49
- Write the answer. The number is 7
Check. 7 × 7 = 49. ✓
Watch out. You are looking for the number that is multiplied by itself, not half of 49.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. Try the numbers 1, 2, 3, … in order. Work out each square until you reach 100.
- Write the square numbers in order. Stop when you reach the number you want.1² = 1 2² = 4 3² = 9 4² = 16
5² = 25 6² = 36 7² = 49 8² = 64
9² = 81 10² = 100 - Find 100 in the list. 10² = 100
- Write the answer. The number is 10
Check. 10 × 10 = 100. ✓
Watch out. You are looking for the number that is multiplied by itself, not half of 100.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. Try the numbers 1, 2, 3, … in order. Work out each square until you reach 144.
- Write the square numbers in order. Stop when you reach the number you want.1² = 1 2² = 4 3² = 9 4² = 16
5² = 25 6² = 36 7² = 49 8² = 64
9² = 81 10² = 100 11² = 121 12² = 144 - Find 144 in the list. 12² = 144
- Write the answer. The number is 12
Check. 12 × 12 = 144. ✓
Watch out. You are looking for the number that is multiplied by itself, not half of 144.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. Try the numbers 1, 2, 3, … in order. Work out each square until you reach 81.
- Write the square numbers in order. Stop when you reach the number you want.1² = 1 2² = 4 3² = 9 4² = 16
5² = 25 6² = 36 7² = 49 8² = 64
9² = 81 - Find 81 in the list. 9² = 81
- Write the answer. The number is 9
Check. 9 × 9 = 81. ✓
Watch out. You are looking for the number that is multiplied by itself, not half of 81.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. 13² means 13 multiplied by itself: 13 × 13.
- Say what the little 2 means. The little 2 means “multiply the number by itself”. It does not mean “times 2”.13² = 13 × 13
- Split the second 13 into 10 and 3. Multiply 13 by each part. This is called partitioning.13 × 13 = 13 × 10 + 13 × 3
13 × 10 = 130
13 × 3 = 39 - Add the two answers. 130 + 39 = 169
Check. 10² = 100 and 20² = 400, and 169 is between them. ✓
Watch out. 13² is not 13 × 2 = 26. Multiply 13 by 13.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. 15² means 15 multiplied by itself: 15 × 15.
- Say what the little 2 means. The little 2 means “multiply the number by itself”. It does not mean “times 2”.15² = 15 × 15
- Split the second 15 into 10 and 5. Multiply 15 by each part. This is called partitioning.15 × 15 = 15 × 10 + 15 × 5
15 × 10 = 150
15 × 5 = 75 - Add the two answers. 150 + 75 = 225
Check. Pattern for numbers ending in 5: 1 × 2 = 2, then write 25 on the end: 225. ✓
Watch out. 15² is not 15 × 2 = 30. Multiply 15 by 15.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. 20² means 20 multiplied by itself: 20 × 20.
- Say what the little 2 means. The little 2 means “multiply the number by itself”. It does not mean “times 2”.20² = 20 × 20
- Split each number into a single digit and a 10. 20 is 2 × 10. Multiplying can be done in any order.20 × 20 = (2 × 10) × (2 × 10) = (2 × 2) × (10 × 10)
- Multiply. 2 × 2 = 4 and 10 × 10 = 100.4 × 100 = 400
Check. 20 × 2 = 40, and 40 × 10 = 400. ✓
Watch out. 20² is not 20 × 2 = 40. Multiply 20 by 20.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. 25² means 25 multiplied by itself: 25 × 25.
- Say what the little 2 means. The little 2 means “multiply the number by itself”. It does not mean “times 2”.25² = 25 × 25
- Split the second 25 into 20 and 5. Multiply 25 by each part. This is called partitioning.25 × 25 = 25 × 20 + 25 × 5
25 × 20 = 500
25 × 5 = 125 - Add the two answers. 500 + 125 = 625
Check. Pattern for numbers ending in 5: 2 × 3 = 6, then write 25 on the end: 625. ✓
Watch out. 25² is not 25 × 2 = 50. Multiply 25 by 25.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. 30² means 30 multiplied by itself: 30 × 30.
- Say what the little 2 means. The little 2 means “multiply the number by itself”. It does not mean “times 2”.30² = 30 × 30
- Split each number into a single digit and a 10. 30 is 3 × 10. Multiplying can be done in any order.30 × 30 = (3 × 10) × (3 × 10) = (3 × 3) × (10 × 10)
- Multiply. 3 × 3 = 9 and 10 × 10 = 100.9 × 100 = 900
Check. 30 × 3 = 90, and 90 × 10 = 900. ✓
Watch out. 30² is not 30 × 2 = 60. Multiply 30 by 30.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. 40² means 40 multiplied by itself: 40 × 40.
- Say what the little 2 means. The little 2 means “multiply the number by itself”. It does not mean “times 2”.40² = 40 × 40
- Split each number into a single digit and a 10. 40 is 4 × 10. Multiplying can be done in any order.40 × 40 = (4 × 10) × (4 × 10) = (4 × 4) × (10 × 10)
- Multiply. 4 × 4 = 16 and 10 × 10 = 100.16 × 100 = 1,600
Check. 40 × 4 = 160, and 160 × 10 = 1,600. ✓
Watch out. 40² is not 40 × 2 = 80. Multiply 40 by 40.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. 50² means 50 multiplied by itself: 50 × 50.
- Say what the little 2 means. The little 2 means “multiply the number by itself”. It does not mean “times 2”.50² = 50 × 50
- Split each number into a single digit and a 10. 50 is 5 × 10. Multiplying can be done in any order.50 × 50 = (5 × 10) × (5 × 10) = (5 × 5) × (10 × 10)
- Multiply. 5 × 5 = 25 and 10 × 10 = 100.25 × 100 = 2,500
Check. 50 × 5 = 250, and 250 × 10 = 2,500. ✓
Watch out. 50² is not 50 × 2 = 100. Multiply 50 by 50.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. 80² means 80 multiplied by itself: 80 × 80.
- Say what the little 2 means. The little 2 means “multiply the number by itself”. It does not mean “times 2”.80² = 80 × 80
- Split each number into a single digit and a 10. 80 is 8 × 10. Multiplying can be done in any order.80 × 80 = (8 × 10) × (8 × 10) = (8 × 8) × (10 × 10)
- Multiply. 8 × 8 = 64 and 10 × 10 = 100.64 × 100 = 6,400
Check. 80 × 8 = 640, and 640 × 10 = 6,400. ✓
Watch out. 80² is not 80 × 2 = 160. Multiply 80 by 80.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. Write the square numbers in order, and see if 36 is one of them.
- List the square numbers. Keep going until you reach or pass 36.1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25, 6² = 36
- Look for the number. 36 is in the list.
- Answer the question. 36 is a square number: Yes
Check. 6 × 6 = 36. ✓
Watch out. Do not guess from the last digit. Check the list of square numbers.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. Write the square numbers in order, and see if 50 is one of them.
- List the square numbers. Keep going until you reach or pass 50.1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25, 6² = 36, 7² = 49, 8² = 64
- Look for the number. 50 is between 49 and 64. It is not in the list.
- Answer the question. 50 is not a square number: No
Check. No whole number multiplied by itself makes 50. ✓
Watch out. Do not guess from the last digit. Check the list of square numbers.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. Write the square numbers in order, and see if 81 is one of them.
- List the square numbers. Keep going until you reach or pass 81.1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25, 6² = 36, 7² = 49, 8² = 64, 9² = 81
- Look for the number. 81 is in the list.
- Answer the question. 81 is a square number: Yes
Check. 9 × 9 = 81. ✓
Watch out. Do not guess from the last digit. Check the list of square numbers.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. Write the square numbers in order, and see if 90 is one of them.
- List the square numbers. Keep going until you reach or pass 90.1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25, 6² = 36, 7² = 49, 8² = 64, 9² = 81, 10² = 100
- Look for the number. 90 is between 81 and 100. It is not in the list.
- Answer the question. 90 is not a square number: No
Check. No whole number multiplied by itself makes 90. ✓
Watch out. Do not guess from the last digit. Check the list of square numbers.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. Write the square numbers in order, and see if 121 is one of them.
- List the square numbers. Keep going until you reach or pass 121.1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25, 6² = 36, 7² = 49, 8² = 64, 9² = 81, 10² = 100, 11² = 121
- Look for the number. 121 is in the list.
- Answer the question. 121 is a square number: Yes
Check. 11 × 11 = 121. ✓
Watch out. Do not guess from the last digit. Check the list of square numbers.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. Write the square numbers in order, and see if 200 is one of them.
- List the square numbers. Keep going until you reach or pass 200.1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25, 6² = 36, 7² = 49, 8² = 64, 9² = 81, 10² = 100, 11² = 121, 12² = 144, 13² = 169, 14² = 196, 15² = 225
- Look for the number. 200 is between 196 and 225. It is not in the list.
- Answer the question. 200 is not a square number: No
Check. No whole number multiplied by itself makes 200. ✓
Watch out. Do not guess from the last digit. Check the list of square numbers.
1, 4, ?, 16, ?, 36, 49, ?, 81, ?
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. The square numbers are 1², 2², 3², … Work out the square number for each missing place.
- Number the places. The number in place 3 is 3², and so on.1², 2², 3², 4², 5², 6², 7², 8², 9², 10²
- Work out each missing number. Multiply the place number by itself.Place 3: 3² = 3 × 3 = 9
Place 5: 5² = 5 × 5 = 25
Place 8: 8² = 8 × 8 = 64
Place 10: 10² = 10 × 10 = 100 - Write the missing numbers. 9, 25, 64, 100
Check. All together: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. ✓
Watch out. Do not add the same number each time. The gaps between square numbers get bigger.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. 13² means 13 multiplied by itself: 13 × 13.
- Say what the little 2 means. The little 2 means “multiply the number by itself”. It does not mean “times 2”.13² = 13 × 13
- Split the second 13 into 10 and 3. Multiply 13 by each part. This is called partitioning.13 × 13 = 13 × 10 + 13 × 3
13 × 10 = 130
13 × 3 = 39 - Add the two answers. 130 + 39 = 169
Check. 10² = 100 and 20² = 400, and 169 is between them. ✓
Watch out. 13² is not 13 × 2 = 26. Multiply 13 by 13.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. Try the numbers 1, 2, 3, … in order. Work out each square until you reach 144.
- Write the square numbers in order. Stop when you reach the number you want.1² = 1 2² = 4 3² = 9 4² = 16
5² = 25 6² = 36 7² = 49 8² = 64
9² = 81 10² = 100 11² = 121 12² = 144 - Find 144 in the list. 12² = 144
- Write the answer. The number is 12
Check. 12 × 12 = 144. ✓
Watch out. You are looking for the number that is multiplied by itself, not half of 144.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. Work out the correct answer yourself first. Then compare it with the answer in the question to find the mistake.
- Work it out carefully. Do the question again, one step at a time.6² = 6 × 6 = 36
- Compare with the answer in the question. Leo said 6² = 12.
- Find the mistake. 6 × 2 = 12. Leo multiplied by 2. But the little 2 means “multiply 6 by itself”.
- Write the correct answer. 6² = 6 × 6 = 36.
Check. 6 rows of 6 dots: 6, 12, 18, 24, 30, 36. ✓
Watch out. The little 2 is not a multiplier. It says how many times the number is written.
Key words. A counter-example is one example that shows a statement is false.
Method. Try some examples. If the statement is true for every example you can find, ask why. If you find one that is false, it cannot be “always”.
- Try some numbers. Test the statement with some examples.3 + 5 = 8 7 + 9 = 16 11 + 13 = 24 1 + 1 = 2
- Explain why. A reason shows what is happening for every number, not just the ones we tried.An odd number is a pair-up number with one left over: 7 = 3 pairs + 1. Two odd numbers each have 1 left over, and those two make another pair. So there is nothing left over, and the answer is even.
- Decide. Always
Check. Odd numbers end in 1, 3, 5, 7 or 9, and two of them add to a number ending in 0, 2, 4, 6 or 8. ✓
Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not. “Never” means no example works.
Key words. A counter-example is one example that shows a statement is false.
Method. Try some examples. If the statement is true for every example you can find, ask why. If you find one that is false, it cannot be “always”.
- Try some numbers. Test the statement with some examples.4 = 4 × 1 ✓ 16 = 4 × 4 ✓ 36 = 4 × 9 ✓
- Look for one that does not work. One example that breaks the rule is called a counter-example.9 ÷ 4 = 2 remainder 1, so 9 is a square number but not a multiple of 4. (25 and 49 are the same.)
- Explain why. A reason shows what is happening for every number, not just the ones we tried.Squares of even numbers are multiples of 4 (2 × 2 × (something)). Squares of odd numbers are odd, so they cannot be multiples of 4.
- Decide. Sometimes
Check. 4 is a multiple of 4 and 9 is not, so we have one of each. ✓
Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not. “Never” means no example works.
Key words. A counter-example is one example that shows a statement is false.
Method. Try some examples. If the statement is true for every example you can find, ask why. If you find one that is false, it cannot be “always”.
- Try some numbers. Test the statement with some examples.The first ten squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. The last digits are 1, 4, 9, 6, 5, 6, 9, 4, 1, 0.
- Explain why. A reason shows what is happening for every number, not just the ones we tried.The last digit of a square only depends on the last digit of the number you square: 0² ends in 0, 1² in 1, 2² in 4, 3² in 9, 4² in 6, 5² in 5, 6² in 6, 7² in 9, 8² in 4 and 9² in 1. None of these is 7.
- Decide. Never
Check. Square numbers can only end in 0, 1, 4, 5, 6 or 9. ✓
Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not. “Never” means no example works.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. Work out each side of the equation. If both sides give the same answer, the statement is true.
- Work out the left side. Square each number, then add.3² = 3 × 3 = 9 and 4² = 4 × 4 = 16
9 + 16 = 25 - Work out the right side. 5² = 5 × 5 = 25
- Compare. Both sides are 25.25 = 25, so it is true.
Check. In the dot squares, the 3 by 3 square (9 dots) and the 4 by 4 square (16 dots) together make 25 dots, the same as a 5 by 5 square. ✓
Watch out. Square first, then add. 3² + 4² is not (3 + 4)².
Key words. A clue puzzle gives you facts about a mystery number. Use the clues one at a time to cross out numbers until only one is left. Then check it against every clue.
Method. Use the clues one at a time. Start with the clue that gives the shortest list, then cross out the numbers that break the other clues.
- Start with the clue that gives a short list: “a square number”. List the numbers between 20 and 80 that fit this clue.25, 36, 49, 64
- Use the next clue: “an even number”. Cross out every number that does not fit.
25, 36,49, 64 - Use the next clue: “a multiple of 3”. Cross out every number that does not fit.36,
64 - Check the answer against every clue. It must fit all of them.“a square number” ✓
“an even number” ✓
“a multiple of 3” ✓
Check. Only 36 is left, and it fits every clue. ✓
Watch out. Check the range carefully: “between 30 and 70” does not include 30 or 70.
Exercise 3H Cube numbers
Non-calculatorKey words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.
Method. 1³ means 1 × 1 × 1. Multiply the first two, then multiply the answer by 1 again.
- Say what the little 3 means. The little 3 means “multiply three of the number together”. It does not mean “times 3”.1³ = 1 × 1 × 1
- Multiply the first two numbers. Work from left to right.1 × 1 = 1
- Multiply that answer by 1 again. 1 × 1 = 1
Check. Picture a cube: 1 layers, each with 1 small cubes. 1 × 1 = 1. ✓
Watch out. 1³ is not 1 × 3 = 3. It is 1 × 1 × 1.
Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.
Method. 2³ means 2 × 2 × 2. Multiply the first two, then multiply the answer by 2 again.
- Say what the little 3 means. The little 3 means “multiply three of the number together”. It does not mean “times 3”.2³ = 2 × 2 × 2
- Multiply the first two numbers. Work from left to right.2 × 2 = 4
- Multiply that answer by 2 again. 4 × 2 = 8
Check. Picture a cube: 2 layers, each with 4 small cubes. 2 × 4 = 8. ✓
Watch out. 2³ is not 2 × 3 = 6. It is 2 × 2 × 2.
Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.
Method. 3³ means 3 × 3 × 3. Multiply the first two, then multiply the answer by 3 again.
- Say what the little 3 means. The little 3 means “multiply three of the number together”. It does not mean “times 3”.3³ = 3 × 3 × 3
- Multiply the first two numbers. Work from left to right.3 × 3 = 9
- Multiply that answer by 3 again. 9 × 3 = 27
Check. Picture a cube: 3 layers, each with 9 small cubes. 3 × 9 = 27. ✓
Watch out. 3³ is not 3 × 3 = 9. It is 3 × 3 × 3.
Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.
Method. 4³ means 4 × 4 × 4. Multiply the first two, then multiply the answer by 4 again.
- Say what the little 3 means. The little 3 means “multiply three of the number together”. It does not mean “times 3”.4³ = 4 × 4 × 4
- Multiply the first two numbers. Work from left to right.4 × 4 = 16
- Multiply that answer by 4 again. 16 × 4 = 64
Check. Picture a cube: 4 layers, each with 16 small cubes. 4 × 16 = 64. ✓
Watch out. 4³ is not 4 × 3 = 12. It is 4 × 4 × 4.
Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.
Method. 2³ means 2 × 2 × 2. Multiply the first two, then multiply the answer by 2 again.
- Say what the little 3 means. The little 3 means “multiply three of the number together”. It does not mean “times 3”.2³ = 2 × 2 × 2
- Multiply the first two numbers. Work from left to right.2 × 2 = 4
- Multiply that answer by 2 again. 4 × 2 = 8
Check. Picture a cube: 2 layers, each with 4 small cubes. 2 × 4 = 8. ✓
Watch out. 2³ is not 2 × 3 = 6. It is 2 × 2 × 2.
Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.
Method. 3³ means 3 × 3 × 3. Multiply the first two, then multiply the answer by 3 again.
- Say what the little 3 means. The little 3 means “multiply three of the number together”. It does not mean “times 3”.3³ = 3 × 3 × 3
- Multiply the first two numbers. Work from left to right.3 × 3 = 9
- Multiply that answer by 3 again. 9 × 3 = 27
Check. Picture a cube: 3 layers, each with 9 small cubes. 3 × 9 = 27. ✓
Watch out. 3³ is not 3 × 3 = 9. It is 3 × 3 × 3.
Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.
Method. 5³ means 5 × 5 × 5. Multiply the first two, then multiply the answer by 5 again.
- Say what the little 3 means. The little 3 means “multiply three of the number together”. It does not mean “times 3”.5³ = 5 × 5 × 5
- Multiply the first two numbers. Work from left to right.5 × 5 = 25
- Multiply that answer by 5 again. 25 × 5 = 20 × 5 + 5 × 5 = 100 + 25 = 125
Check. Picture a cube: 5 layers, each with 25 small cubes. 5 × 25 = 125. ✓
Watch out. 5³ is not 5 × 3 = 15. It is 5 × 5 × 5.
Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.
Method. 10³ means 10 × 10 × 10. Multiply the first two, then multiply the answer by 10 again.
- Say what the little 3 means. The little 3 means “multiply three of the number together”. It does not mean “times 3”.10³ = 10 × 10 × 10
- Multiply the first two numbers. Work from left to right.10 × 10 = 100
- Multiply that answer by 10 again. 100 × 10 = 100 × 10 + 0 × 10 = 1000 + 0 = 1,000
Check. Picture a cube: 10 layers, each with 100 small cubes. 10 × 100 = 1,000. ✓
Watch out. 10³ is not 10 × 3 = 30. It is 10 × 10 × 10.
Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.
Method. 6³ means 6 × 6 × 6. Multiply the first two, then multiply the answer by 6 again.
- Say what the little 3 means. The little 3 means “multiply three of the number together”. It does not mean “times 3”.6³ = 6 × 6 × 6
- Multiply the first two numbers. Work from left to right.6 × 6 = 36
- Multiply that answer by 6 again. 36 × 6 = 30 × 6 + 6 × 6 = 180 + 36 = 216
Check. Picture a cube: 6 layers, each with 36 small cubes. 6 × 36 = 216. ✓
Watch out. 6³ is not 6 × 3 = 18. It is 6 × 6 × 6.
Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.
Method. 7³ means 7 × 7 × 7. Multiply the first two, then multiply the answer by 7 again.
- Say what the little 3 means. The little 3 means “multiply three of the number together”. It does not mean “times 3”.7³ = 7 × 7 × 7
- Multiply the first two numbers. Work from left to right.7 × 7 = 49
- Multiply that answer by 7 again. 49 × 7 = 40 × 7 + 9 × 7 = 280 + 63 = 343
Check. Picture a cube: 7 layers, each with 49 small cubes. 7 × 49 = 343. ✓
Watch out. 7³ is not 7 × 3 = 21. It is 7 × 7 × 7.
Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.
Method. 9³ means 9 × 9 × 9. Multiply the first two, then multiply the answer by 9 again.
- Say what the little 3 means. The little 3 means “multiply three of the number together”. It does not mean “times 3”.9³ = 9 × 9 × 9
- Multiply the first two numbers. Work from left to right.9 × 9 = 81
- Multiply that answer by 9 again. 81 × 9 = 80 × 9 + 1 × 9 = 720 + 9 = 729
Check. Picture a cube: 9 layers, each with 81 small cubes. 9 × 81 = 729. ✓
Watch out. 9³ is not 9 × 3 = 27. It is 9 × 9 × 9.
Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.
Method. Write the cube numbers in order, and see if 8 is one of them.
- List the cube numbers. Keep going until you reach or pass 8.1³ = 1, 2³ = 8
- Look for the number. 8 is in the list.
- Answer the question. 8 is a cube number: Yes
Check. 2 × 2 × 2 = 8. ✓
Watch out. Do not guess from the last digit. Check the list of cube numbers.
Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.
Method. Write the cube numbers in order, and see if 20 is one of them.
- List the cube numbers. Keep going until you reach or pass 20.1³ = 1, 2³ = 8, 3³ = 27
- Look for the number. 20 is between 8 and 27. It is not in the list.
- Answer the question. 20 is not a cube number: No
Check. No whole number multiplied by itself three times makes 20. ✓
Watch out. Do not guess from the last digit. Check the list of cube numbers.
Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.
Method. Write the cube numbers in order, and see if 125 is one of them.
- List the cube numbers. Keep going until you reach or pass 125.1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125
- Look for the number. 125 is in the list.
- Answer the question. 125 is a cube number: Yes
Check. 5 × 5 × 5 = 125. ✓
Watch out. Do not guess from the last digit. Check the list of cube numbers.
Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.
Method. Write the cube numbers in order, and see if 100 is one of them.
- List the cube numbers. Keep going until you reach or pass 100.1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125
- Look for the number. 100 is between 64 and 125. It is not in the list.
- Answer the question. 100 is not a cube number: No
Check. No whole number multiplied by itself three times makes 100. ✓
Watch out. Do not guess from the last digit. Check the list of cube numbers.
Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.
Method. Write the cube numbers in order, and see if 343 is one of them.
- List the cube numbers. Keep going until you reach or pass 343.1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125, 6³ = 216, 7³ = 343
- Look for the number. 343 is in the list.
- Answer the question. 343 is a cube number: Yes
Check. 7 × 7 × 7 = 343. ✓
Watch out. Do not guess from the last digit. Check the list of cube numbers.
Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.
Method. Write the cube numbers in order, and see if 1,000 is one of them.
- List the cube numbers. Keep going until you reach or pass 1,000.1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125, 6³ = 216, 7³ = 343, 8³ = 512, 9³ = 729, 10³ = 1,000
- Look for the number. 1,000 is in the list.
- Answer the question. 1,000 is a cube number: Yes
Check. 10 × 10 × 10 = 1,000. ✓
Watch out. Do not guess from the last digit. Check the list of cube numbers.
1, 8, ?, 64, ?, 216, ?, 512
Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.
Method. The cube numbers are 1³, 2³, 3³, … Work out the cube number for each missing place.
- Number the places. The number in place 3 is 3³, and so on.1³, 2³, 3³, 4³, 5³, 6³, 7³, 8³
- Work out each missing number. Multiply the place number by itself twice.Place 3: 3³ = 3 × 3 × 3 = 27
Place 5: 5³ = 5 × 5 × 5 = 125
Place 7: 7³ = 7 × 7 × 7 = 343 - Write the missing numbers. 27, 125, 343
Check. All together: 1, 8, 27, 64, 125, 216, 343, 512. ✓
Watch out. Do not add the same number each time. The gaps between cube numbers get bigger.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. Work out each power first. Then do the addition or subtraction.
- Work out 5². 5² means 5 × 5.5² = 5 × 5 = 25
- Work out 2³. 2³ means 2 × 2 × 2.2³ = 2 × 2 × 2 = 8
- Now subtract. Use the two answers.25 − 8 = 17
Check. Estimate: 25 is about 20 and 8 is about 10, so the answer should be near 10. ✓
Watch out. Do not multiply the number by the little number: 5² is not 5 × 2.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. Work out each power first. Then do the addition or subtraction.
- Work out 4³. 4³ means 4 × 4 × 4.4³ = 4 × 4 × 4 = 64
- Work out 6². 6² means 6 × 6.6² = 6 × 6 = 36
- Now subtract. Use the two answers.64 − 36 = 28
Check. Estimate: 64 is about 60 and 36 is about 40, so the answer should be near 20. ✓
Watch out. Do not multiply the number by the little number: 4³ is not 4 × 3.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. Work out each power first. Then do the addition or subtraction.
- Work out 2³. 2³ means 2 × 2 × 2.2³ = 2 × 2 × 2 = 8
- Work out 4². 4² means 4 × 4.4² = 4 × 4 = 16
- Now add. Use the two answers.8 + 16 = 24
Check. Estimate: 8 is about 10 and 16 is about 20, so the answer should be near 30. ✓
Watch out. Do not multiply the number by the little number: 2³ is not 2 × 3.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. Work out each power first. Then do the addition or subtraction.
- Work out 10³. 10³ means 10 × 10 × 10.10³ = 10 × 10 × 10 = 1,000
- Work out 10². 10² means 10 × 10.10² = 10 × 10 = 100
- Now subtract. Use the two answers.1,000 − 100 = 900
Check. Estimate: 1,000 is about 1000 and 100 is about 100, so the answer should be near 900. ✓
Watch out. Do not multiply the number by the little number: 10³ is not 10 × 3.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”. Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.
Method. Work out 4³ and 8². Then compare them.
- Work out 4³. 4³ means 4 × 4 × 4.4 × 4 = 16, and 16 × 4 = 64
- Work out 8². 8² means 8 × 8.8 × 8 = 64
- Compare. They are the same!4³ = 64 and 8² = 64. They are equal.
Check. 64 is both a cube number (4³) and a square number (8²). ✓
Watch out. Work each one out. Do not guess that a cube is bigger than a square.
Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.
Method. 3³ means 3 × 3 × 3. Multiply the first two, then multiply the answer by 3 again.
- Say what the little 3 means. The little 3 means “multiply three of the number together”. It does not mean “times 3”.3³ = 3 × 3 × 3
- Multiply the first two numbers. Work from left to right.3 × 3 = 9
- Multiply that answer by 3 again. 9 × 3 = 27
Check. Picture a cube: 3 layers, each with 9 small cubes. 3 × 9 = 27. ✓
Watch out. 3³ is not 3 × 3 = 9. It is 3 × 3 × 3.
Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.
Method. A 3 by 3 by 3 cube is made of 27 small cubes. Think about which small cube is hidden in the middle.
- Count all the small cubes. The cube has 3 layers of 9.3 × 3 × 3 = 27
- Find the cube with no paint. Only the small cube in the very middle touches nothing on the outside.1 small cube has no paint.
- Take it away from the total. All the others have paint on at least one face.27 − 1 = 26
Check. Count by layers: top layer 9, bottom layer 9, middle layer 8 (all but the centre). 9 + 9 + 8 = 26. ✓
Watch out. The middle cube is the only one that cannot be seen from outside.
Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.
Method. 10³ means 10 × 10 × 10. Multiply the first two, then multiply the answer by 10 again.
- Say what the little 3 means. The little 3 means “multiply three of the number together”. It does not mean “times 3”.10³ = 10 × 10 × 10
- Multiply the first two numbers. Work from left to right.10 × 10 = 100
- Multiply that answer by 10 again. 100 × 10 = 100 × 10 + 0 × 10 = 1000 + 0 = 1,000
Check. Picture a cube: 10 layers, each with 100 small cubes. 10 × 100 = 1,000. ✓
Watch out. 10³ is not 10 × 3 = 30. It is 10 × 10 × 10.
Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.
Method. Try the numbers 1, 2, 3, … in order. Work out each cube until you reach 125.
- Write the cube numbers in order. Stop when you reach the number you want.1³ = 1 2³ = 8 3³ = 27 4³ = 64
5³ = 125 - Find 125 in the list. 5³ = 125
- Write the answer. The number is 5
Check. 5 × 5 × 5 = 125. ✓
Watch out. You are looking for the number that is multiplied by itself, not half of 125.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”. Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.
Method. List the cube numbers in the range. Then test each one to see if it is also a square number.
- List the cube numbers between 50 and 100. Cube the numbers 1, 2, 3, … until you pass the top of the range.4³ = 64
- Test each one. Is it a square number?64: 8 × 8 = 64: square
- Write the number. 64 (4³ and 8²)
Check. 4 × 4 × 4 = 64 and 8 × 8 = 64. ✓
Watch out. A number that is both a square and a cube must pass both tests.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”. Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.
Method. List the cube numbers in the range. Then test each one to see if it is also a square number.
- List the cube numbers between 700 and 800. Cube the numbers 1, 2, 3, … until you pass the top of the range.9³ = 729
- Test each one. Is it a square number?729: 27 × 27 = 729: square
- Write the number. 729 (9³ and 27²)
Check. 9 × 9 × 9 = 729 and 27 × 27 = 729. ✓
Watch out. A number that is both a square and a cube must pass both tests.