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This page sets out the method for powers & roots in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Powers & roots topic page.
When you see a small raised number, multiply the base by itself that many times.
When you see or , ask 'which number multiplied by itself (or by itself twice) gives this?'
A square number makes a square array. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
5² means 5 × 5 = 25. The power is 2.
A cube uses the factor three times. 2³ = 2 × 2 × 2 = 8.
The power tells you how many factors, not what you multiply by.
10³ = 10 × 10 × 10 = 1 000. Three factors of 10.
Anything to the power 1 is itself. Anything non-zero to the power 0 is 1, which can wait until later.
√49 asks “which positive number times itself is 49?”. The answer is 7, because 7² = 49.
A square root undoes a square.
If you know 6² = 36, then 7² = 36 + 6 + 7 = 49. You add the next strip of the square.
That check stops you memorising a wrong square.
“Square”, “to the power 2” and “squared” mean the same here.
“Square root” is the inverse question. Do not halve the number.
The first squares are worth knowing by heart because roots and area questions lean on them: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. Those are 1^2 up to 15^2. If a square slips, rebuild it. 15^2 is 15 × 10 = 150 and 15 × 5 = 75, and 150 + 75 = 225.
A square grows faster than the double. 12^2 = 144, while 12 × 2 = 24. If an answer is only a little bigger than the base, the index was used as a factor instead of a count. For a base bigger than 1, the square is bigger than the base itself.
You can step from a known square to the next one. 13^2 = 169. One way to see the jump from 12^2 is 144 + 12 + 13 = 169: you add the side you had and the new side. That check is useful when 169 and 196 feel easy to swap. 14^2 = 196 and 15^2 = 225.
Method: 1. A square is the number multiplied by itself. 2. Split a larger square into tens and units if you do not know it yet. 3. Add those parts to get the square.
The cubes that come up again and again are 1^3 = 1, 2^3 = 8, 3^3 = 27, 4^3 = 64, 5^3 = 125 and 10^3 = 1000. Say them as repeated factors. 3^3 = 3 × 3 × 3. First 3 × 3 = 9, then 9 × 3 = 27. Stopping after two factors gives the square, not the cube.
10^3 = 10 × 10 × 10 = 1000. Three factors of 10 put three zeros on the end. 10^2 = 100 has two zeros. The index and the number of zeros match for powers of 10, which is a useful check and a poor habit to copy onto other bases. 2^3 is not 2000 and it is not 6.
A cube of a whole number above 1 is larger than the square of the same number. 5^2 = 25 and 5^3 = 125. If a volume is given as 125 cubic centimetres and the solid is a cube, the edge is the cube root, 5 cm, not the square root. Read whether the power in the story is an area or a volume.
Method: 1. A cube uses the number as a factor three times. 2. Square it first, then multiply by the number again. 3. Two factors make the square. The cube needs a third factor.
Square root asks a backwards question: which positive number times itself gives this? √(144) = 12 because 12 × 12 = 144. Halving is a different question. Half of 144 is 72, and 72 × 72 is nothing like 144. The root sign is not a division by 2.
Cube root asks which number used three times as a factor gives this. The cube root of 8 is 2, of 27 is 3, of 64 is 4, of 125 is 5 and of 1000 is 10. If you recognise the cube, you can write the root down. If you do not, test a small whole number by cubing it.
In a mixed line, a root is an index-level operation, so it happens before multiplication, addition or subtraction. 8^2 + 3^3 - √(121) starts with 64, 27 and 11. Only then do you add and subtract: 64 + 27 = 91 and 91 - 11 = 80. Do not subtract 121 and then try to take a root of the whole line.
Method: 1. A square root asks which positive number multiplies by itself to give this. 2. A cube root asks which number used three times gives this. 3. Check by squaring or cubing your answer.
When the number under a square root is not a square you know, find the two squares it sits between. For √(40), list 6^2 = 36 and 7^2 = 49. Because 36 < 40 < 49, the root sits between 6 and 7. You are not asked for a decimal unless the question says so. The two whole numbers are the answer.
Say which end it is nearer only if the question asks. 40 - 36 = 4 and 49 - 40 = 9, so 40 is closer to 36 than to 49, and √(40) is closer to 6 than to 7. That comparison is extra. The basic fact is the pair of integers.
The same idea works for a cube that is not in the list. 30 is between 27 = 3^3 and 64 = 4^3, so the cube root of 30 is between 3 and 4. Stay with squares unless the question says cube. Mixing the two lists is how √(64) gets answered as 4 when the question wanted the square root, which is 8, or the other way round.
Method: 1. Find the two square numbers the value sits between. 2. The root sits between those two whole numbers. 3. Do not give a decimal unless the question asks for one.
A power tells you how many copies of the base to multiply together. In 2^5 the base is 2 and the power (or index) is 5, so 2^5 = 2 × 2 × 2 × 2 × 2 = 32. It does not mean 2 × 5. Exam questions use small bases like 2, 3, 5 and 10, so work up one multiplication at a time: 2, 4, 8, 16, 32.
The reverse also appears: "Write 3 × 3 × 3 × 3 as a power of 3". Count the threes. There are four, so the answer is 3^4, which equals 81. Powers of ten are the easiest to work out, because 10^4 is a 1 followed by four zeros, which is 10 000. Learn the table of small powers: 2^6 = 64, 3^4 = 81, 5^3 = 125. It saves a lot of time.
Method: 1. Write the base the number of times shown by the power. 2. Multiply two at a time, writing the running totals. 3. Check that you have used the base exactly as many times as the power says. 4. For powers of ten, count the zeros instead of multiplying.
Tip: 2^5 is not 10. If your answer is the base times the power, go back and start again.
Many Foundation questions give a list of numbers and ask for a square number, a cube number or a multiple. Know your squares to 15^2 = 225 and your cubes 1, 8, 27, 64, 125, 216 by heart. Phrases such as "one less than a cube number" mean you look for a number that becomes a cube when you add one. 26 + 1 = 27 = 3^3, so 26 is one less than a cube number.
A range question such as "Write down all the cube numbers between 10 and 200" needs a list. Check the first cube above the range and the last one below it: 2^3 = 8 is too small, then 27, 64, 125 are in range and 216 is too big. Write the full list so the examiner can see you did not miss one. A common trap is to forget that 1 is both a square number and a cube number.
Method: 1. Write out the list of squares or cubes that are near the range. 2. Cross out the ones outside the range given in the question. 3. For neighbours, add or subtract the amount in the question and compare with your list. 4. Write all the answers asked for, in order.
If a number is not a perfect square, its square root is a decimal. For √(60), find the two squares either side of 60. Since 7^2 = 49 and 8^2 = 64, the root lies between 7 and 8. Because 60 is close to 64, the root is nearer to 8. The question may then say "show that the root is between 7 and 8", which means you must write both squares.
The same idea works for cube roots. sqrt[3]100 lies between 4 and 5, because 4^3 = 64 and 5^3 = 125. Squares of multiples of ten and of decimals follow place value: 30^2 = 900, 0.3^2 = 0.09, √(400) = 20 and √(0.25) = 0.5. Check any root by squaring the answer and getting back to the number you started with.
Method: 1. Write down the nearest square or cube number below the value. 2. Write down the nearest one above. 3. State that the value lies between their roots, using the two whole numbers. 4. Square or cube the answer to check it.
Tip: For decimals, count the places: 0.3^2 has two decimal places, so 0.3^2 = 0.09.
Exam lines such as 5^3 - √(169) or 4^3 - 6^2 + √(81) mix powers and roots. Work out each power and root first, then do the addition and subtraction from left to right. In the first, 5^3 = 125 and √(169) = 13, so the answer is 112. Write the values under the original line so each step is visible.
Be careful with a negative sign next to a power and with a power after a bracket. If you see 2 + 3^2, square before adding: 2 + 9 = 11, not 25. A context version may ask for the side of a cube with volume 343 cm^3, which is the cube root, 7 cm. Then the surface area is six faces of 7 × 7. Write the units, cm^2 for area, to finish the question properly.
Method: 1. Evaluate every power and every root first. 2. Replace each one in the line by its value. 3. Do multiplication and division, then addition and subtraction. 4. Check units: area needs squared units and volume needs cubed units.
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