lesson
A root reverses a power
1 · Learn
Cubing means multiplying three equal factors: 3³ = 3 × 3 × 3 = 27. The cube root asks which number cubes to 27, so the cube root of 27 is 3. A model cube with three unit cubes along each edge has three layers of nine unit cubes: 27 altogether.
Odd powers keep a negative sign: (-2)³ = -8, so the cube root of -8 is -2. Even powers behave differently: 3⁴ = 81 and (-3)⁴ = 81. The principal fourth root of 81 is the non-negative answer 3, but the equation x⁴ = 81 has two real solutions, x = 3 and x = -3.
A root need not be a whole number. Since 1² < 2 < 2², sqrt(2) is between 1 and 2. Its exact value is sqrt(2); approximately 1.41 to two decimal places. Squaring 1.41 gives 1.9881, not exactly 2. Keep exact root notation when an exact answer is requested.
Check the root index: a cube root reverses a third power and a fourth root reverses a fourth power. A square-root button does not find every kind of root. A negative number has no real even root, because an even power of any real number is non-negative.
2 · Worked example
The cube root of 64 is 4 because 4 × 4 × 4 = 64. The principal fourth root of 16 is 2, but x⁴ = 16 has solutions 2 and -2. Separately, sqrt(10) lies between 3 and 4 because 9 < 10 < 16.
3 · Your turn
Find the cube root of -125 and the principal fourth root of 81. Give both real solutions of x⁴ = 81. Between which consecutive integers does sqrt(7) lie?
Check your answer
-5; 3; x = 3 or -3. sqrt(7) is between 2 and 3 because 4 < 7 < 9. Its exact form is sqrt(7), not a rounded decimal.
4 · Apply your learning
Build or sketch the three layers of a 3-by-3-by-3 cube. Count cubes in one layer and then all layers. Explain why the edge length is a cube root of the total count, not its square root.
Adult guidance and safety
Ask pupils to justify classifications and operations, not just repeat a rule. Use a number line and exact fractions before decimal approximations. Check the scope of a negative sign and distinguish a principal root from all solutions of an equation.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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