GCSE Maths: Estimating roots and powers, and using a calculator (Foundation)
Part of the GCSE Maths factors, multiples and primes chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Estimating roots and powers, and using a calculator
Most numbers are not perfect squares. For example 30 is not a square number, so √30 is not a whole number. But we can still say roughly how big it is. Look at the squares either side of 30: 52 = 25 and 62 = 36. Since 25 < 30 < 36, the square root of 30 is between 5 and 6.
Which whole number is it nearer to? The halfway number is 5.5, and 5.52 = 30.25. Because 30 is a little less than 30.25, √30 is a little less than 5.5, so it is nearer to 5.
Estimating to 1 decimal place (trial and improvement). Square numbers with one decimal place between 5 and 6: 5.42 = 29.16 (too small), 5.52 = 30.25 (too big). So √30 is between 5.4 and 5.5. To choose, test the halfway value 5.45: 5.452 = 29.7025, which is less than 30, so √30 is above 5.45 and rounds to 5.5.
Cube roots work the same way with cubes: 33 = 27 and 43 = 64, so the cube root of 50 is between 3 and 4.
Estimating powers. Round the base to an easy number first. 3.92 is close to 42 = 16. Because 3.9 is slightly less than 4, the true answer is slightly less than 16 (it is 15.21).
Using a calculator. Use the x2 and x3 keys for squares and cubes, the power key (x□ or ^) for other powers, and the √ key for roots (on many calculators you press √ first, then the number, then =). Use brackets for negative numbers and for working under a root: √(6.42 − 3.22) needs brackets round the whole of the working under the root. Always estimate first, to catch key-pressing mistakes.
Key ideas
- Roots of non-square numbers lie between two whole numbers: find the squares (or cubes) either side.
- To decide which is nearer, test the halfway number.
- To get 1 d.p., square the one-decimal-place numbers (trial and improvement) and test the halfway value.
- Estimate a power by rounding the base first.
- On a calculator use the power and root keys, use brackets, and check with an estimate.
Key words. To estimate is to find an answer that is close, without finding it exactly. A number that is not a perfect square or cube has a root that is a decimal.
Method. Find the two whole numbers whose squares are either side of 30.
- Find the square numbers just below and just above 30. Use the square numbers you know.52 = 25
62 = 36
25 < 30 < 36 - Take the square root of each part of the inequality. Taking a square root keeps the order, so the root of 30 is between 5 and 6.5 < √30 < 6
- Decide which whole number is nearer. Halfway between 5 and 6 is 5.5, and 5.52 = 30.25. If 30 is below 30.25 the root is below halfway.30 is less than 30.25, so √30 is nearer to 5.
Check. Use a calculator: √30 = 5.477..., which is between 5 and 6 and nearer to 5. ✓
Watch out. √30 is not a whole number here, so do not give an exact whole-number answer.
Key words. Trial and improvement means trying a value, seeing if it is too big or too small, and adjusting it. 1 decimal place (1 d.p.) means one digit after the decimal point.
Method. Use trial and improvement: square one-decimal-place numbers until you have trapped 30 between two of them.
- Find the whole numbers either side. Use the square numbers you know.52 = 25 and 62 = 36, so 5 < √30 < 6
- Try one-decimal-place numbers between them, using the position of 30 as a guide. 30 is closer to 25 than to 36, which tells us where to start. Work out the square of each trial.5.42 = 29.16
5.52 = 30.25
5.62 = 31.36 - Find the two one-decimal-place numbers that 30 lies between. We want one trial answer that is too small and the next one that is too big.5.42 = 29.16 < 30 < 30.25 = 5.52
- Test the halfway value to decide which one-decimal-place number is nearer. Halfway between 5.4 and 5.5 is 5.45. If 30 is bigger than 5.452 the root is bigger than 5.45 and rounds up.5.452 = 29.7025
30 is more than 29.7025, so √30 is above 5.45. - Write the answer to 1 decimal place. √30 ≈ 5.5
Check. A calculator gives √30 = 5.4772..., which rounds to 5.5 to 1 decimal place. ✓
Watch out. To round correctly, always test the halfway value. Do not just pick the trial answer that looks closer.
Key words. ≈ means “approximately equal to”.
Method. To estimate a power, round the base first (to the nearest whole number or 1 significant figure), then work out the power.
- Round 3.9 to a whole number. A whole number is easier to square or cube in your head.3.9 ≈ 4
- Work out the power of the rounded number. Use the rounded value in place of the original.42 = 16
- Say it is an estimate. Use the ≈ sign, which means “is approximately equal to”.3.92 ≈ 16
Check. The exact value is 15.21, which is close to 16. Because 3.9 is smaller than 4, the exact answer is smaller than the estimate. ✓
Watch out. Round the base before squaring or cubing, not the answer.
Key words. d.p. means decimal places and s.f. means significant figures.
Method. Use the power key on the calculator, then round as asked.
- Enter the number and use the power key. Your calculator has a key that raises a number to a power. Enter the base first.Press 1.7, then the power key, then 4, then =.
- Read the display. The calculator shows the full answer.1.74 = 8.3521
- Round to 2 decimal places. Look at the next digit: 5 or more rounds up.1.74 = 8.35 to 2 decimal places
Check. Estimate: round 1.7 to 2 and work out 24 = 16. This is close to 8.35. ✓
Watch out. Use brackets for a negative base: (−3)2 is 9 but −32 is −9.
Key words. d.p. means decimal places.
Method. Use the root key on the calculator, then round as asked.
- Use the √ key. Your calculator finds roots with a special key. On many calculators you press the root key first, then enter the number, then =.√77 = 8.774964...
- Round to 2 decimal places. Look at the digit after the last one you keep: 5 or more rounds up.√77 = 8.77 to 2 d.p.
- Check that the answer is sensible. Find the square numbers either side of 77: the root must be between the two whole numbers.82 = 64 < 77 < 81 = 92, so the root is between 8 and 9. ✓
Check. Check by raising to the power: 8.772 = 76.9129, which is close to 77. ✓
Watch out. A calculator answer is usually a long decimal. Do not write the whole display: round as asked.
Watch out
Do not give a calculator answer with all its digits unless the question asks for it: round as asked. And a root that is not a whole number cannot be found exactly as a decimal: 5.48 is only an approximation of √30, so we use the sign ≈.Exercise 2I Estimating roots and powers (non-calculator)
Non-calculatorKey words. To estimate is to find an answer that is close, without finding it exactly. A number that is not a perfect square or cube has a root that is a decimal.
Method. Find the two whole numbers whose squares are either side of 20.
- Find the square numbers just below and just above 20. Use the square numbers you know.42 = 16
52 = 25
16 < 20 < 25 - Take the square root of each part of the inequality. Taking a square root keeps the order, so the root of 20 is between 4 and 5.4 < √20 < 5
Check. Use a calculator: √20 = 4.472..., which is between 4 and 5. ✓
Watch out. √20 is not a whole number here, so do not give an exact whole-number answer.
Key words. To estimate is to find an answer that is close, without finding it exactly. A number that is not a perfect square or cube has a root that is a decimal.
Method. Find the two whole numbers whose squares are either side of 50.
- Find the square numbers just below and just above 50. Use the square numbers you know.72 = 49
82 = 64
49 < 50 < 64 - Take the square root of each part of the inequality. Taking a square root keeps the order, so the root of 50 is between 7 and 8.7 < √50 < 8
Check. Use a calculator: √50 = 7.071..., which is between 7 and 8. ✓
Watch out. √50 is not a whole number here, so do not give an exact whole-number answer.
Key words. To estimate is to find an answer that is close, without finding it exactly. A number that is not a perfect square or cube has a root that is a decimal.
Method. Find the two whole numbers whose squares are either side of 75.
- Find the square numbers just below and just above 75. Use the square numbers you know.82 = 64
92 = 81
64 < 75 < 81 - Take the square root of each part of the inequality. Taking a square root keeps the order, so the root of 75 is between 8 and 9.8 < √75 < 9
Check. Use a calculator: √75 = 8.660..., which is between 8 and 9. ✓
Watch out. √75 is not a whole number here, so do not give an exact whole-number answer.
Key words. To estimate is to find an answer that is close, without finding it exactly. A number that is not a perfect square or cube has a root that is a decimal.
Method. Find the two whole numbers whose squares are either side of 110.
- Find the square numbers just below and just above 110. Use the square numbers you know.102 = 100
112 = 121
100 < 110 < 121 - Take the square root of each part of the inequality. Taking a square root keeps the order, so the root of 110 is between 10 and 11.10 < √110 < 11
Check. Use a calculator: √110 = 10.488..., which is between 10 and 11. ✓
Watch out. √110 is not a whole number here, so do not give an exact whole-number answer.
Key words. To estimate is to find an answer that is close, without finding it exactly. A number that is not a perfect square or cube has a root that is a decimal.
Method. Find the two whole numbers whose squares are either side of 3.
- Find the square numbers just below and just above 3. Use the square numbers you know.12 = 1
22 = 4
1 < 3 < 4 - Take the square root of each part of the inequality. Taking a square root keeps the order, so the root of 3 is between 1 and 2.1 < √3 < 2
Check. Use a calculator: √3 = 1.732..., which is between 1 and 2. ✓
Watch out. √3 is not a whole number here, so do not give an exact whole-number answer.
Key words. To estimate is to find an answer that is close, without finding it exactly. A number that is not a perfect square or cube has a root that is a decimal.
Method. Find the two whole numbers whose squares are either side of 200.
- Find the square numbers just below and just above 200. Use the square numbers you know.142 = 196
152 = 225
196 < 200 < 225 - Take the square root of each part of the inequality. Taking a square root keeps the order, so the root of 200 is between 14 and 15.14 < √200 < 15
Check. Use a calculator: √200 = 14.142..., which is between 14 and 15. ✓
Watch out. √200 is not a whole number here, so do not give an exact whole-number answer.
Key words. To estimate is to find an answer that is close, without finding it exactly. A number that is not a perfect square or cube has a root that is a decimal.
Method. Find the two whole numbers whose squares are either side of 10.
- Find the square numbers just below and just above 10. Use the square numbers you know.32 = 9
42 = 16
9 < 10 < 16 - Take the square root of each part of the inequality. Taking a square root keeps the order, so the root of 10 is between 3 and 4.3 < √10 < 4
- Decide which whole number is nearer. Halfway between 3 and 4 is 3.5, and 3.52 = 12.25. If 10 is below 12.25 the root is below halfway.10 is less than 12.25, so √10 is nearer to 3.
Check. Use a calculator: √10 = 3.162..., which is between 3 and 4 and nearer to 3. ✓
Watch out. √10 is not a whole number here, so do not give an exact whole-number answer.
Key words. To estimate is to find an answer that is close, without finding it exactly. A number that is not a perfect square or cube has a root that is a decimal.
Method. Find the two whole numbers whose squares are either side of 20.
- Find the square numbers just below and just above 20. Use the square numbers you know.42 = 16
52 = 25
16 < 20 < 25 - Take the square root of each part of the inequality. Taking a square root keeps the order, so the root of 20 is between 4 and 5.4 < √20 < 5
- Decide which whole number is nearer. Halfway between 4 and 5 is 4.5, and 4.52 = 20.25. If 20 is below 20.25 the root is below halfway.20 is less than 20.25, so √20 is nearer to 4.
Check. Use a calculator: √20 = 4.472..., which is between 4 and 5 and nearer to 4. ✓
Watch out. √20 is not a whole number here, so do not give an exact whole-number answer.
Key words. To estimate is to find an answer that is close, without finding it exactly. A number that is not a perfect square or cube has a root that is a decimal.
Method. Find the two whole numbers whose squares are either side of 30.
- Find the square numbers just below and just above 30. Use the square numbers you know.52 = 25
62 = 36
25 < 30 < 36 - Take the square root of each part of the inequality. Taking a square root keeps the order, so the root of 30 is between 5 and 6.5 < √30 < 6
- Decide which whole number is nearer. Halfway between 5 and 6 is 5.5, and 5.52 = 30.25. If 30 is below 30.25 the root is below halfway.30 is less than 30.25, so √30 is nearer to 5.
Check. Use a calculator: √30 = 5.477..., which is between 5 and 6 and nearer to 5. ✓
Watch out. √30 is not a whole number here, so do not give an exact whole-number answer.
Key words. To estimate is to find an answer that is close, without finding it exactly. A number that is not a perfect square or cube has a root that is a decimal.
Method. Find the two whole numbers whose squares are either side of 40.
- Find the square numbers just below and just above 40. Use the square numbers you know.62 = 36
72 = 49
36 < 40 < 49 - Take the square root of each part of the inequality. Taking a square root keeps the order, so the root of 40 is between 6 and 7.6 < √40 < 7
- Decide which whole number is nearer. Halfway between 6 and 7 is 6.5, and 6.52 = 42.25. If 40 is below 42.25 the root is below halfway.40 is less than 42.25, so √40 is nearer to 6.
Check. Use a calculator: √40 = 6.325..., which is between 6 and 7 and nearer to 6. ✓
Watch out. √40 is not a whole number here, so do not give an exact whole-number answer.
Key words. To estimate is to find an answer that is close, without finding it exactly. A number that is not a perfect square or cube has a root that is a decimal.
Method. Find the two whole numbers whose squares are either side of 55.
- Find the square numbers just below and just above 55. Use the square numbers you know.72 = 49
82 = 64
49 < 55 < 64 - Take the square root of each part of the inequality. Taking a square root keeps the order, so the root of 55 is between 7 and 8.7 < √55 < 8
- Decide which whole number is nearer. Halfway between 7 and 8 is 7.5, and 7.52 = 56.25. If 55 is below 56.25 the root is below halfway.55 is less than 56.25, so √55 is nearer to 7.
Check. Use a calculator: √55 = 7.416..., which is between 7 and 8 and nearer to 7. ✓
Watch out. √55 is not a whole number here, so do not give an exact whole-number answer.
Key words. To estimate is to find an answer that is close, without finding it exactly. A number that is not a perfect square or cube has a root that is a decimal.
Method. Find the two whole numbers whose squares are either side of 90.
- Find the square numbers just below and just above 90. Use the square numbers you know.92 = 81
102 = 100
81 < 90 < 100 - Take the square root of each part of the inequality. Taking a square root keeps the order, so the root of 90 is between 9 and 10.9 < √90 < 10
- Decide which whole number is nearer. Halfway between 9 and 10 is 9.5, and 9.52 = 90.25. If 90 is below 90.25 the root is below halfway.90 is less than 90.25, so √90 is nearer to 9.
Check. Use a calculator: √90 = 9.487..., which is between 9 and 10 and nearer to 9. ✓
Watch out. √90 is not a whole number here, so do not give an exact whole-number answer.
Key words. To estimate is to find an answer that is close, without finding it exactly. A number that is not a perfect square or cube has a root that is a decimal.
Method. Find the two whole numbers whose cubes are either side of 10.
- Find the cube numbers just below and just above 10. Use the cube numbers you know.23 = 8
33 = 27
8 < 10 < 27 - Take the cube root of each part of the inequality. Taking a cube root keeps the order, so the root of 10 is between 2 and 3.2 < 3√10 < 3
Check. Use a calculator: 3√10 = 2.154..., which is between 2 and 3. ✓
Watch out. 3√10 is not a whole number here, so do not give an exact whole-number answer.
Key words. To estimate is to find an answer that is close, without finding it exactly. A number that is not a perfect square or cube has a root that is a decimal.
Method. Find the two whole numbers whose cubes are either side of 30.
- Find the cube numbers just below and just above 30. Use the cube numbers you know.33 = 27
43 = 64
27 < 30 < 64 - Take the cube root of each part of the inequality. Taking a cube root keeps the order, so the root of 30 is between 3 and 4.3 < 3√30 < 4
Check. Use a calculator: 3√30 = 3.107..., which is between 3 and 4. ✓
Watch out. 3√30 is not a whole number here, so do not give an exact whole-number answer.
Key words. To estimate is to find an answer that is close, without finding it exactly. A number that is not a perfect square or cube has a root that is a decimal.
Method. Find the two whole numbers whose cubes are either side of 70.
- Find the cube numbers just below and just above 70. Use the cube numbers you know.43 = 64
53 = 125
64 < 70 < 125 - Take the cube root of each part of the inequality. Taking a cube root keeps the order, so the root of 70 is between 4 and 5.4 < 3√70 < 5
Check. Use a calculator: 3√70 = 4.121..., which is between 4 and 5. ✓
Watch out. 3√70 is not a whole number here, so do not give an exact whole-number answer.
Key words. To estimate is to find an answer that is close, without finding it exactly. A number that is not a perfect square or cube has a root that is a decimal.
Method. Find the two whole numbers whose cubes are either side of 200.
- Find the cube numbers just below and just above 200. Use the cube numbers you know.53 = 125
63 = 216
125 < 200 < 216 - Take the cube root of each part of the inequality. Taking a cube root keeps the order, so the root of 200 is between 5 and 6.5 < 3√200 < 6
Check. Use a calculator: 3√200 = 5.848..., which is between 5 and 6. ✓
Watch out. 3√200 is not a whole number here, so do not give an exact whole-number answer.
Key words. Trial and improvement means trying a value, seeing if it is too big or too small, and adjusting it. 1 decimal place (1 d.p.) means one digit after the decimal point.
Method. Use trial and improvement: square one-decimal-place numbers until you have trapped 10 between two of them.
- Find the whole numbers either side. Use the square numbers you know.32 = 9 and 42 = 16, so 3 < √10 < 4
- Try one-decimal-place numbers between them, using the position of 10 as a guide. 10 is closer to 9 than to 16, which tells us where to start. Work out the square of each trial.3.12 = 9.61
3.22 = 10.24 - Find the two one-decimal-place numbers that 10 lies between. We want one trial answer that is too small and the next one that is too big.3.12 = 9.61 < 10 < 10.24 = 3.22
- Test the halfway value to decide which one-decimal-place number is nearer. Halfway between 3.1 and 3.2 is 3.15. If 10 is bigger than 3.152 the root is bigger than 3.15 and rounds up.3.152 = 9.9225
10 is more than 9.9225, so √10 is above 3.15. - Write the answer to 1 decimal place. √10 ≈ 3.2
Check. A calculator gives √10 = 3.1623..., which rounds to 3.2 to 1 decimal place. ✓
Watch out. To round correctly, always test the halfway value. Do not just pick the trial answer that looks closer.
Key words. Trial and improvement means trying a value, seeing if it is too big or too small, and adjusting it. 1 decimal place (1 d.p.) means one digit after the decimal point.
Method. Use trial and improvement: square one-decimal-place numbers until you have trapped 20 between two of them.
- Find the whole numbers either side. Use the square numbers you know.42 = 16 and 52 = 25, so 4 < √20 < 5
- Try one-decimal-place numbers between them, using the position of 20 as a guide. 20 is closer to 16 than to 25, which tells us where to start. Work out the square of each trial.4.42 = 19.36
4.52 = 20.25 - Find the two one-decimal-place numbers that 20 lies between. We want one trial answer that is too small and the next one that is too big.4.42 = 19.36 < 20 < 20.25 = 4.52
- Test the halfway value to decide which one-decimal-place number is nearer. Halfway between 4.4 and 4.5 is 4.45. If 20 is bigger than 4.452 the root is bigger than 4.45 and rounds up.4.452 = 19.8025
20 is more than 19.8025, so √20 is above 4.45. - Write the answer to 1 decimal place. √20 ≈ 4.5
Check. A calculator gives √20 = 4.4721..., which rounds to 4.5 to 1 decimal place. ✓
Watch out. To round correctly, always test the halfway value. Do not just pick the trial answer that looks closer.
Key words. Trial and improvement means trying a value, seeing if it is too big or too small, and adjusting it. 1 decimal place (1 d.p.) means one digit after the decimal point.
Method. Use trial and improvement: square one-decimal-place numbers until you have trapped 30 between two of them.
- Find the whole numbers either side. Use the square numbers you know.52 = 25 and 62 = 36, so 5 < √30 < 6
- Try one-decimal-place numbers between them, using the position of 30 as a guide. 30 is closer to 25 than to 36, which tells us where to start. Work out the square of each trial.5.42 = 29.16
5.52 = 30.25
5.62 = 31.36 - Find the two one-decimal-place numbers that 30 lies between. We want one trial answer that is too small and the next one that is too big.5.42 = 29.16 < 30 < 30.25 = 5.52
- Test the halfway value to decide which one-decimal-place number is nearer. Halfway between 5.4 and 5.5 is 5.45. If 30 is bigger than 5.452 the root is bigger than 5.45 and rounds up.5.452 = 29.7025
30 is more than 29.7025, so √30 is above 5.45. - Write the answer to 1 decimal place. √30 ≈ 5.5
Check. A calculator gives √30 = 5.4772..., which rounds to 5.5 to 1 decimal place. ✓
Watch out. To round correctly, always test the halfway value. Do not just pick the trial answer that looks closer.
Key words. Trial and improvement means trying a value, seeing if it is too big or too small, and adjusting it. 1 decimal place (1 d.p.) means one digit after the decimal point.
Method. Use trial and improvement: square one-decimal-place numbers until you have trapped 50 between two of them.
- Find the whole numbers either side. Use the square numbers you know.72 = 49 and 82 = 64, so 7 < √50 < 8
- Try one-decimal-place numbers between them, using the position of 50 as a guide. 50 is closer to 49 than to 64, which tells us where to start. Work out the square of each trial.72 = 49
7.12 = 50.41
7.22 = 51.84 - Find the two one-decimal-place numbers that 50 lies between. We want one trial answer that is too small and the next one that is too big.72 = 49 < 50 < 50.41 = 7.12
- Test the halfway value to decide which one-decimal-place number is nearer. Halfway between 7 and 7.1 is 7.05. If 50 is bigger than 7.052 the root is bigger than 7.05 and rounds up.7.052 = 49.7025
50 is more than 49.7025, so √50 is above 7.05. - Write the answer to 1 decimal place. √50 ≈ 7.1
Check. A calculator gives √50 = 7.0711..., which rounds to 7.1 to 1 decimal place. ✓
Watch out. To round correctly, always test the halfway value. Do not just pick the trial answer that looks closer.
Key words. Trial and improvement means trying a value, seeing if it is too big or too small, and adjusting it. 1 decimal place (1 d.p.) means one digit after the decimal point.
Method. Use trial and improvement: square one-decimal-place numbers until you have trapped 75 between two of them.
- Find the whole numbers either side. Use the square numbers you know.82 = 64 and 92 = 81, so 8 < √75 < 9
- Try one-decimal-place numbers between them, using the position of 75 as a guide. 75 is further from 64 than to 81, which tells us where to start. Work out the square of each trial.8.62 = 73.96
8.72 = 75.69 - Find the two one-decimal-place numbers that 75 lies between. We want one trial answer that is too small and the next one that is too big.8.62 = 73.96 < 75 < 75.69 = 8.72
- Test the halfway value to decide which one-decimal-place number is nearer. Halfway between 8.6 and 8.7 is 8.65. If 75 is bigger than 8.652 the root is bigger than 8.65 and rounds up.8.652 = 74.8225
75 is more than 74.8225, so √75 is above 8.65. - Write the answer to 1 decimal place. √75 ≈ 8.7
Check. A calculator gives √75 = 8.6603..., which rounds to 8.7 to 1 decimal place. ✓
Watch out. To round correctly, always test the halfway value. Do not just pick the trial answer that looks closer.
Key words. Trial and improvement means trying a value, seeing if it is too big or too small, and adjusting it. 1 decimal place (1 d.p.) means one digit after the decimal point.
Method. Use trial and improvement: square one-decimal-place numbers until you have trapped 12 between two of them.
- Find the whole numbers either side. Use the square numbers you know.32 = 9 and 42 = 16, so 3 < √12 < 4
- Try one-decimal-place numbers between them, using the position of 12 as a guide. 12 is closer to 9 than to 16, which tells us where to start. Work out the square of each trial.3.42 = 11.56
3.52 = 12.25 - Find the two one-decimal-place numbers that 12 lies between. We want one trial answer that is too small and the next one that is too big.3.42 = 11.56 < 12 < 12.25 = 3.52
- Test the halfway value to decide which one-decimal-place number is nearer. Halfway between 3.4 and 3.5 is 3.45. If 12 is bigger than 3.452 the root is bigger than 3.45 and rounds up.3.452 = 11.9025
12 is more than 11.9025, so √12 is above 3.45. - Write the answer to 1 decimal place. √12 ≈ 3.5
Check. A calculator gives √12 = 3.4641..., which rounds to 3.5 to 1 decimal place. ✓
Watch out. To round correctly, always test the halfway value. Do not just pick the trial answer that looks closer.
Key words. ≈ means “approximately equal to”.
Method. To estimate a power, round the base first (to the nearest whole number or 1 significant figure), then work out the power.
- Round 2.9 to a whole number. A whole number is easier to square or cube in your head.2.9 ≈ 3
- Work out the power of the rounded number. Use the rounded value in place of the original.32 = 9
- Say it is an estimate. Use the ≈ sign, which means “is approximately equal to”.2.92 ≈ 9
Check. The exact value is 8.41, which is close to 9. Because 2.9 is smaller than 3, the exact answer is smaller than the estimate. ✓
Watch out. Round the base before squaring or cubing, not the answer.
Key words. ≈ means “approximately equal to”.
Method. To estimate a power, round the base first (to the nearest whole number or 1 significant figure), then work out the power.
- Round 4.1 to a whole number. A whole number is easier to square or cube in your head.4.1 ≈ 4
- Work out the power of the rounded number. Use the rounded value in place of the original.42 = 16
- Say it is an estimate. Use the ≈ sign, which means “is approximately equal to”.4.12 ≈ 16
Check. The exact value is 16.81, which is close to 16. Because 4.1 is bigger than 4, the exact answer is bigger than the estimate. ✓
Watch out. Round the base before squaring or cubing, not the answer.
Key words. ≈ means “approximately equal to”.
Method. To estimate a power, round the base first (to the nearest whole number or 1 significant figure), then work out the power.
- Round 6.8 to a whole number. A whole number is easier to square or cube in your head.6.8 ≈ 7
- Work out the power of the rounded number. Use the rounded value in place of the original.72 = 49
- Say it is an estimate. Use the ≈ sign, which means “is approximately equal to”.6.82 ≈ 49
Check. The exact value is 46.24, which is close to 49. Because 6.8 is smaller than 7, the exact answer is smaller than the estimate. ✓
Watch out. Round the base before squaring or cubing, not the answer.
Key words. ≈ means “approximately equal to”.
Method. To estimate a power, round the base first (to the nearest whole number or 1 significant figure), then work out the power.
- Round 3.1 to a whole number. A whole number is easier to square or cube in your head.3.1 ≈ 3
- Work out the power of the rounded number. Use the rounded value in place of the original.33 = 27
- Say it is an estimate. Use the ≈ sign, which means “is approximately equal to”.3.13 ≈ 27
Check. The exact value is 29.791, which is close to 27. Because 3.1 is bigger than 3, the exact answer is bigger than the estimate. ✓
Watch out. Round the base before squaring or cubing, not the answer.
Key words. ≈ means “approximately equal to”.
Method. To estimate a power, round the base first (to the nearest whole number or 1 significant figure), then work out the power.
- Round 4.9 to a whole number. A whole number is easier to square or cube in your head.4.9 ≈ 5
- Work out the power of the rounded number. Use the rounded value in place of the original.52 = 25
- Say it is an estimate. Use the ≈ sign, which means “is approximately equal to”.4.92 ≈ 25
Check. The exact value is 24.01, which is close to 25. Because 4.9 is smaller than 5, the exact answer is smaller than the estimate. ✓
Watch out. Round the base before squaring or cubing, not the answer.
Key words. ≈ means “approximately equal to”.
Method. To estimate a power, round the base first (to the nearest whole number or 1 significant figure), then work out the power.
- Round 1.9 to a whole number. A whole number is easier to square or cube in your head.1.9 ≈ 2
- Work out the power of the rounded number. Use the rounded value in place of the original.23 = 8
- Say it is an estimate. Use the ≈ sign, which means “is approximately equal to”.1.93 ≈ 8
Check. The exact value is 6.859, which is close to 8. Because 1.9 is smaller than 2, the exact answer is smaller than the estimate. ✓
Watch out. Round the base before squaring or cubing, not the answer.
Key words. Trial and improvement means trying a value, seeing if it is too big or too small, and adjusting it. 1 decimal place (1 d.p.) means one digit after the decimal point.
Method. Use trial and improvement: square one-decimal-place numbers until you have trapped 70 between two of them.
- Find the whole numbers either side. Use the square numbers you know.82 = 64 and 92 = 81, so 8 < √70 < 9
- Try one-decimal-place numbers between them, using the position of 70 as a guide. 70 is closer to 64 than to 81, which tells us where to start. Work out the square of each trial.8.32 = 68.89
8.42 = 70.56
8.52 = 72.25 - Find the two one-decimal-place numbers that 70 lies between. We want one trial answer that is too small and the next one that is too big.8.32 = 68.89 < 70 < 70.56 = 8.42
- Test the halfway value to decide which one-decimal-place number is nearer. Halfway between 8.3 and 8.4 is 8.35. If 70 is bigger than 8.352 the root is bigger than 8.35 and rounds up.8.352 = 69.7225
70 is more than 69.7225, so √70 is above 8.35. - Write the answer to 1 decimal place. √70 ≈ 8.4
Check. A calculator gives √70 = 8.3666..., which rounds to 8.4 to 1 decimal place. ✓
Watch out. To round correctly, always test the halfway value. Do not just pick the trial answer that looks closer.
Key words. To square a number, multiply it by itself (52 = 5 × 5 = 25). To cube a number, multiply three of it (53 = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.
Method. Work out the number under the root first. Then find the squares either side of it.
- Work out the number under the root. The root sign works like a bracket: do the working inside first.32 + 52 = 9 + 25 = 34
- Find the squares either side of 34. Use the square numbers you know.52 = 25 and 62 = 36, so 5 < √34 < 6
- Decide which is nearer. Halfway between 5 and 6 is 5.5, and 5.52 = 30.25. Since 34 is more than 30.25, √34 is more than 5.5.√34 is nearer to 6.
Check. √34 = 5.83..., which is nearest to 6. ✓
Watch out. Do not take the roots separately: √(9 + 25) is not 3 + 5.
Key words. To square a number, multiply it by itself (52 = 5 × 5 = 25). To cube a number, multiply three of it (53 = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.
Method. Compare 50 with the squares of 7 and 8.
- Work out the two squares. 72 = 49 and 82 = 64
- Compare with 50. How far is 50 from each square?50 − 49 = 1 and 64 − 50 = 14
- Use the halfway number. Halfway between 7 and 8 is 7.5 and 7.52 = 56.25. Since 50 < 56.25, √50 is below 7.5.√50 is between 7 and 7.5, so nearer to 7.
Check. √50 = 7.07..., which is nearer to 7. ✓
Key words. To square a number, multiply it by itself (52 = 5 × 5 = 25). To cube a number, multiply three of it (53 = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.
Method. Square 6.5 and compare the answer with 40.
- Work out 6.52. Multiply 6.5 by itself: 65 × 65 = 4225, then put in two decimal places.6.52 = 42.25
- Compare with 40. If 6.52 is bigger than 40, then 6.5 is bigger than the square root of 40.42.25 > 40
- Conclude. Squaring keeps the order for positive numbers.√40 < 6.5
Check. √40 = 6.32..., which is less than 6.5. ✓
Key words. To square a number, multiply it by itself (52 = 5 × 5 = 25). To cube a number, multiply three of it (53 = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.
Method. Test Alex’s answer by squaring it.
- Square Alex’s answer. If √30 = 15 then 152 must be 30.152 = 15 × 15 = 225
- Compare. 225 is far bigger than 30.225 ≠ 30, so Alex is wrong.
- Find the correct size. Look at the squares either side of 30.52 = 25 and 62 = 36, so √30 is between 5 and 6
Check. 5.52 = 30.25, which is close to 30. ✓
Watch out. A square root is not half. It is the number that multiplies by itself to give the number.
Key words. To square a number, multiply it by itself (52 = 5 × 5 = 25). To cube a number, multiply three of it (53 = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.
Method. Round each number to an easy value (a square number for the root, a whole number for the power).
- Round inside the root. 49.3 is close to 49, which is 72.√49.3 ≈ √49 = 7
- Round the base of the power. 2.9 is close to 3.2.92 ≈ 32 = 9
- Add. Add the two estimates.7 + 9 = 16
Check. A calculator gives 7.02 + 8.41 = 15.43, which is close to 16. ✓
Watch out. An estimate is not exact, so use the sign ≈.
Key words. To square a number, multiply it by itself (52 = 5 × 5 = 25). To cube a number, multiply three of it (53 = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.
Method. Square 6.5 and 6.6. Then find the whole number between the two squares.
- Square 6.5. 65 × 65 = 4225, then two decimal places.6.52 = 42.25
- Square 6.6. 66 × 66 = 4356, then two decimal places.6.62 = 43.56
- Find the whole number in between. We need 42.25 < n < 43.56.n = 43
Check. √43 = 6.557..., which is between 6.5 and 6.6. ✓
Watch out. There is only one whole number between 42.25 and 43.56.
Exercise 2J Powers and roots with a calculator
Calculator allowedKey words. d.p. means decimal places and s.f. means significant figures.
Method. Use the power key on the calculator, then round as asked.
- Enter the number and use the power key. Your calculator has a key that raises a number to a power. Enter the base first.Press 1.7, then the power key, then 4, then =.
- Read the display. The calculator shows the full answer.1.74 = 8.3521
- Round to 2 decimal places. Look at the next digit: 5 or more rounds up.1.74 = 8.35 to 2 decimal places
Check. Estimate: round 1.7 to 2 and work out 24 = 16. This is close to 8.35. ✓
Watch out. Use brackets for a negative base: (−3)2 is 9 but −32 is −9.
Key words. d.p. means decimal places and s.f. means significant figures.
Method. Use the power key on the calculator, then round as asked.
- Enter the number and use the x3 key. Your calculator has a special key for squares and cubes. Enter the base first.Press 2.3, then x3, then =.
- Read the display. The calculator shows the full answer.2.33 = 12.167
- Round to 1 decimal place. Look at the next digit: 5 or more rounds up.2.33 = 12.2 to 1 decimal place
Check. Estimate: round 2.3 to 2 and work out 23 = 8. This is close to 12.2. ✓
Watch out. Use brackets for a negative base: (−3)2 is 9 but −32 is −9.
Key words. d.p. means decimal places and s.f. means significant figures.
Method. Use the power key on the calculator, then round as asked.
- Enter the number and use the x2 key. Your calculator has a special key for squares and cubes. Enter the base first.Press 0.85, then x2, then =.
- Read the display. The calculator shows the full answer.0.852 = 0.7225
- Round to 3 decimal places. Look at the next digit: 5 or more rounds up.0.852 = 0.723 to 3 decimal places
Check. Estimate: round 0.85 to 1 and work out 12 = 1. This is close to 0.723. ✓
Watch out. Use brackets for a negative base: (−3)2 is 9 but −32 is −9.
Key words. d.p. means decimal places and s.f. means significant figures.
Method. Use the power key on the calculator, then round as asked.
- Enter the number and use the x3 key. Your calculator has a special key for squares and cubes. Enter the base first.Press 12.5, then x3, then =.
- Read the display. The calculator shows the full answer.12.53 = 1953.125
- Round to 3 significant figures. Look at the next digit: 5 or more rounds up.12.53 = 1950 to 3 significant figures
Check. Estimate: round 12.5 to 13 and work out 133 = 2197. This is close to 1950. ✓
Watch out. Use brackets for a negative base: (−3)2 is 9 but −32 is −9.
Key words. d.p. means decimal places and s.f. means significant figures.
Method. Use the power key on the calculator, then round as asked.
- Enter the number and use the power key. Your calculator has a key that raises a number to a power. Enter the base first.Press 3.2, then the power key, then 5, then =.
- Read the display. The calculator shows the full answer.3.25 = 335.54432
- Round to 1 decimal place. Look at the next digit: 5 or more rounds up.3.25 = 335.5 to 1 decimal place
Check. Estimate: round 3.2 to 3 and work out 35 = 243. This is close to 335.5. ✓
Watch out. Use brackets for a negative base: (−3)2 is 9 but −32 is −9.
Key words. d.p. means decimal places and s.f. means significant figures.
Method. Use the power key on the calculator, then round as asked.
- Enter the number and use the power key. Your calculator has a key that raises a number to a power. Enter the base first.Press 1.05, then the power key, then 10, then =.
- Read the display. The calculator shows the full answer.1.0510 = 1.62889462677744140625
- Round to 3 decimal places. Look at the next digit: 5 or more rounds up.1.0510 = 1.629 to 3 decimal places
Check. Estimate: round 1.05 to 1 and work out 110 = 1. This is close to 1.629. ✓
Watch out. Use brackets for a negative base: (−3)2 is 9 but −32 is −9.
Key words. d.p. means decimal places.
Method. Use the root key on the calculator, then round as asked.
- Use the √ key. Your calculator finds roots with a special key. On many calculators you press the root key first, then enter the number, then =.√77 = 8.774964...
- Round to 2 decimal places. Look at the digit after the last one you keep: 5 or more rounds up.√77 = 8.77 to 2 d.p.
- Check that the answer is sensible. Find the square numbers either side of 77: the root must be between the two whole numbers.82 = 64 < 77 < 81 = 92, so the root is between 8 and 9. ✓
Check. Check by raising to the power: 8.772 = 76.9129, which is close to 77. ✓
Watch out. A calculator answer is usually a long decimal. Do not write the whole display: round as asked.
Key words. d.p. means decimal places.
Method. Use the root key on the calculator, then round as asked.
- Use the √ key. Your calculator finds roots with a special key. On many calculators you press the root key first, then enter the number, then =.√150 = 12.247449...
- Round to 2 decimal places. Look at the digit after the last one you keep: 5 or more rounds up.√150 = 12.25 to 2 d.p.
- Check that the answer is sensible. Find the square numbers either side of 150: the root must be between the two whole numbers.122 = 144 < 150 < 169 = 132, so the root is between 12 and 13. ✓
Check. Check by raising to the power: 12.252 = 150.0625, which is close to 150. ✓
Watch out. A calculator answer is usually a long decimal. Do not write the whole display: round as asked.
Key words. d.p. means decimal places.
Method. Use the root key on the calculator, then round as asked.
- Use the cube root key (3√ ). Your calculator finds roots with a special key. On many calculators you press the root key first, then enter the number, then =.3√250 = 6.299605...
- Round to 1 decimal place. Look at the digit after the last one you keep: 5 or more rounds up.3√250 = 6.3 to 1 d.p.
- Check that the answer is sensible. Find the cube numbers either side of 250: the root must be between the two whole numbers.63 = 216 < 250 < 343 = 73, so the root is between 6 and 7. ✓
Check. Check by raising to the power: 6.33 = 250.047, which is close to 250. ✓
Watch out. A calculator answer is usually a long decimal. Do not write the whole display: round as asked.
Key words. d.p. means decimal places.
Method. Use the root key on the calculator, then round as asked.
- Use the √ key. Your calculator finds roots with a special key. On many calculators you press the root key first, then enter the number, then =.√2 = 1.414214...
- Round to 3 decimal places. Look at the digit after the last one you keep: 5 or more rounds up.√2 = 1.414 to 3 d.p.
- Check that the answer is sensible. Find the square numbers either side of 2: the root must be between the two whole numbers.12 = 1 < 2 < 4 = 22, so the root is between 1 and 2. ✓
Check. Check by raising to the power: 1.4142 = 1.999396, which is close to 2. ✓
Watch out. A calculator answer is usually a long decimal. Do not write the whole display: round as asked.
Key words. d.p. means decimal places.
Method. Use the root key on the calculator, then round as asked.
- Use the cube root key (3√ ). Your calculator finds roots with a special key. On many calculators you press the root key first, then enter the number, then =.3√90 = 4.481405...
- Round to 2 decimal places. Look at the digit after the last one you keep: 5 or more rounds up.3√90 = 4.48 to 2 d.p.
- Check that the answer is sensible. Find the cube numbers either side of 90: the root must be between the two whole numbers.43 = 64 < 90 < 125 = 53, so the root is between 4 and 5. ✓
Check. Check by raising to the power: 4.483 = 89.915392, which is close to 90. ✓
Watch out. A calculator answer is usually a long decimal. Do not write the whole display: round as asked.
Key words. d.p. means decimal places.
Method. Use the root key on the calculator, then round as asked.
- Use the √ key. Your calculator finds roots with a special key. On many calculators you press the root key first, then enter the number, then =.√1000 = 31.622777...
- Round to 1 decimal place. Look at the digit after the last one you keep: 5 or more rounds up.√1000 = 31.6 to 1 d.p.
- Check that the answer is sensible. Find the square numbers either side of 1000: the root must be between the two whole numbers.312 = 961 < 1000 < 1024 = 322, so the root is between 31 and 32. ✓
Check. Check by raising to the power: 31.62 = 998.56, which is close to 1000. ✓
Watch out. A calculator answer is usually a long decimal. Do not write the whole display: round as asked.
Key words. d.p. means decimal places.
Method. Enter the calculation carefully, then round the answer as asked.
- Type the calculation into the calculator, using brackets where needed. Put brackets round 2.5 + 1.3 so that the addition is done before squaring.(2.5 + 1.3)2
- Read the display. The calculator shows many digits.Display: 14.440000...
- Round to 1 decimal place. Look at the next digit: 5 or more rounds up, 4 or less rounds down.= 14.4
Check. Estimate: (2.5 + 1.3) is about 4 and 42 = 16. This is close to 14.4. ✓
Watch out. Use brackets so that the calculator does the working in the right order. Check the answer is sensible with an estimate.
Key words. d.p. means decimal places.
Method. Enter the calculation carefully, then round the answer as asked.
- Type the calculation into the calculator, using brackets where needed. Put brackets round the whole of the working under the root sign: the calculator must do 6.42 − 3.22 = 30.72 before the square root.√(6.42 − 3.22)
- Read the display. The calculator shows many digits.Display: 5.542563...
- Round to 2 decimal places. Look at the next digit: 5 or more rounds up, 4 or less rounds down.= 5.54
Check. Estimate: 62 − 32 = 27 and √27 is about 5. This is close to 5.54. ✓
Watch out. Use brackets so that the calculator does the working in the right order. Check the answer is sensible with an estimate.
Key words. d.p. means decimal places.
Method. Enter the calculation carefully, then round the answer as asked.
- Type the calculation into the calculator, using brackets where needed. First work out 2.53 + 1.53 = 15.625 + 3.375 = 19, then use the cube root key (put the working in brackets).cube root of (2.53 + 1.53) = cube root of 19
- Read the display. The calculator shows many digits.Display: 2.668402...
- Round to 2 decimal places. Look at the next digit: 5 or more rounds up, 4 or less rounds down.= 2.67
Check. Estimate: 2.53 is about 16 and 1.53 about 3, so about 19, and the cube root of 19 is between 2 and 3. This is close to 2.67. ✓
Watch out. Use brackets so that the calculator does the working in the right order. Check the answer is sensible with an estimate.
Key words. d.p. means decimal places.
Method. Enter the calculation carefully, then round the answer as asked.
- Type the calculation into the calculator, using brackets where needed. The power is done before the multiplication, so no brackets are needed.1.23 × 4.5
- Read the display. The calculator shows many digits.Display: 7.776000...
- Round to 2 decimal places. Look at the next digit: 5 or more rounds up, 4 or less rounds down.= 7.78
Check. Estimate: 1.23 is about 2 and 2 × 4.5 = 9. This is close to 7.78. ✓
Watch out. Use brackets so that the calculator does the working in the right order. Check the answer is sensible with an estimate.
Key words. d.p. means decimal places.
Method. Use the root key on the calculator, then round as asked.
- Use the √ key. Your calculator finds roots with a special key. On many calculators you press the root key first, then enter the number, then =.√250 = 15.811388...
- Round to 1 decimal place. Look at the digit after the last one you keep: 5 or more rounds up.√250 = 15.8 to 1 d.p.
- Check that the answer is sensible. Find the square numbers either side of 250: the root must be between the two whole numbers.152 = 225 < 250 < 256 = 162, so the root is between 15 and 16. ✓
Check. Check by raising to the power: 15.82 = 249.64, which is close to 250. ✓
Watch out. A calculator answer is usually a long decimal. Do not write the whole display: round as asked.
Key words. d.p. means decimal places.
Method. Use the root key on the calculator, then round as asked.
- Use the cube root key (3√ ). Your calculator finds roots with a special key. On many calculators you press the root key first, then enter the number, then =.3√500 = 7.937005...
- Round to 1 decimal place. Look at the digit after the last one you keep: 5 or more rounds up.3√500 = 7.9 to 1 d.p.
- Check that the answer is sensible. Find the cube numbers either side of 500: the root must be between the two whole numbers.73 = 343 < 500 < 512 = 83, so the root is between 7 and 8. ✓
Check. Check by raising to the power: 7.93 = 493.039, which is close to 500. ✓
Watch out. A calculator answer is usually a long decimal. Do not write the whole display: round as asked.
Key words. d.p. means decimal places and s.f. means significant figures.
Method. Use the power key on the calculator, then round as asked.
- Enter the number and use the power key. Your calculator has a key that raises a number to a power. Enter the base first.Press 2, then the power key, then 24, then =.
- Read the display. The calculator shows the full answer.224 = 16777216
- Round to 0 decimal places. Look at the next digit: 5 or more rounds up.224 = 16777216 to 0 decimal places
Check. Estimate: round 2 to 2 and work out 224 = 16777216. This is close to 16777216. ✓
Watch out. Use brackets for a negative base: (−3)2 is 9 but −32 is −9.
Key words. To square a number, multiply it by itself (52 = 5 × 5 = 25). To cube a number, multiply three of it (53 = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.
Method. Compare the two calculations carefully, in BIDMAS order.
- Zoe: −42. There is no bracket, so only the 4 is squared. The minus sign is applied after the power.42 = 16, so −42 = −16
- Max: (−4)2. The bracket means the whole of −4 is squared.(−4) × (−4) = 16
- Conclude. They are different calculations, so both are right.Both are correct. Brackets decide what is squared.
Check. Entering (−4)2 with brackets on a calculator gives 16. ✓
Key words. In 25, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 25 means five 2s multiplied together.
Method. Use trial and improvement with the power key. Try values of n and see when the answer first passes 1000.
- Make a first guess. 1.510 is about 57.7, so n is much bigger than 10. Try 15.1.515 = 437.9...
- Try larger values. Increase n one at a time near the target.1.515 = 437.9 < 1000
1.516 = 656.8 < 1000
1.517 = 985.3 < 1000
1.518 = 1477.9 > 1000 - Find the first value over 1000. 1.517 is still under 1000 but 1.518 is over.n = 18
Check. 1.517 = 985.3 < 1000 < 1477.9 = 1.518. ✓
Watch out. “Smallest” means check that the number before it does not work.