Positive integer powers and roots

Edexcel asks for fluent recall of small powers and roots on Foundation Paper 1F (non-calculator) and uses them as building blocks throughout Higher.

Recall facts (must be instant on Paper 1F)

• Squares to 15: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225.
• Cubes to 5: 1, 8, 27, 64, 125. (Cube of 10 = 1000.)
• Square roots: √1 = 1, √4 = 2, … √225 = 15.
• Cube roots: ∛1 = 1, ∛8 = 2, ∛27 = 3, ∛64 = 4, ∛125 = 5, ∛1000 = 10.

Notation

• a² = a × a (read "a squared").
• a³ = a × a × a (read "a cubed").
• aⁿ = a × a × … × a (n factors).
• √a = the positive square root of a.
• ∛a = the cube root of a (which is unique and signed: ∛(−8) = −2).

Powers of negative numbers

• (−3)² = 9 (positive — even power).
• (−3)³ = −27 (negative — odd power).
• −3² = −9 (no brackets — minus is outside; square first).

Indices in equations (Foundation level)

Solve x² = 49 → x = ±7 (always two solutions, even though √49 = 7 is just the positive root). Solve x³ = −64 → x = −4 (unique).

Common Edexcel exam tip

On Paper 1F, the question "find the value of √81 + ∛8" is worth 2 marks. Each correct root scores B1. Always show both intermediate values: 9 + 2 = 11.