Squares, cubes, square roots and cube roots of integers, the crucial difference between (−3)2 and −32, and how to run BODMAS through a calculation containing negatives — worked examples, worksheet with memo and a quiz.
Once the four operations are solid, the exam starts combining them: powers, roots and long mixed calculations. Everything here rests on the sign rules from the previous two lessons, plus careful BODMAS.
1Squares and cubes of integers
Squaring means multiplying a number by itself; cubing means using it three times. The sign follows straight from the multiplication rule: an even power gives a positive answer, an odd power keeps the negative.
- 1(−5)2 = (−5) × (−5) = 25.Two negatives multiplied → positive.
- 2(−2)3 = (−2) × (−2) × (−2) = −8.Three negatives (odd) → negative.
2The bracket trap: (−3)2 versus −32
These two look almost identical and mean completely different things. This single distinction appears in exams every year.
- 1(−3)2 = (−3) × (−3) = 9.The bracket means the whole of −3 is squared.
- 2−32 = −(3 × 3) = −9.With no bracket only the 3 is squared, and the minus is applied afterwards.
Watch the brackets. (−3)² = 9 but −3² = −9. If the negative is inside the bracket it gets squared; if it is outside, it does not.
3Square roots and cube roots
A square root asks: what number times itself gives this? A cube root asks: what number used three times gives this? Note the key difference — you cannot take the square root of a negative number (no number times itself is negative), but you can take the cube root of one.
- 149 = 7, because 7 × 7 = 49.The square root symbol asks for the positive root.
- 227 = 3, because 3 × 3 × 3 = 27.
- 3−8 = −2, because (−2) × (−2) × (−2) = −8.Cube roots of negatives do exist — the odd power keeps the sign.
4Mixed operations — BODMAS with negatives
BODMAS still rules: Brackets, Orders (powers and roots), Divide and Multiply (left to right), then Add and Subtract (left to right). Do one step per line and the signs look after themselves.
- 1Multiplication before addition: (−3) × 4 = −12.BODMAS — × comes before +.
- 2Now −5 + (−12).Rewrite the line with the part you worked out.
- 3Same signs, so add the sizes: −17.
- 1Orders first: (−3)2 = 9.Powers come before × in BODMAS.
- 2Then multiply: 4 × (−2) = −8.
- 3Now 9 − (−8) = 9 + 8 = 17.Subtracting a negative becomes adding.
- 1Brackets first: −12 + 4 = −8.Different signs → 12−4 = 8, bigger size was negative.
- 2Orders: 25 = 5.
- 3Divide: (−8) ÷ (−2) = 4.Same signs → positive.
- 4Finally add: 4 + 5 = 9.
Practice exercises
Work each one out, then click to reveal the answer.
- 1(−5)2
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(−5)(−5) = 25 (even power → positive) - 2(−2)3
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(−2)(−2)(−2) = −8 (odd power stays negative) - 3−42
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−(4 × 4) = −16 (no bracket, so only the 4 is squared) - 4(−4)2
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(−4)(−4) = 16 - 564
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8 - 6−27
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−3, because (−3)(−3)(−3) = −27 - 7−10 + (−2) × 5
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× first: (−2)×5 = −10 → −10 + (−10) = −20 - 8(−3)2 − 4 × (−2)
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9 − (−8) = 9 + 8 = 17 - 9(−20 + 8) ÷ (−3)
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Brackets: −12; then (−12) ÷ (−3) = 4 - 1036 + (−2)3
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6 + (−8) = −2
Quick Quiz
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