Knowledge Organiser: Powers and Roots
Key Terms
- Power/index: How many times to multiply a number by itself
- Base: The number being multiplied (e.g. in 5³, the base is 5)
- Square number: Result of raising a number to the power 2
- Cube number: Result of raising a number to the power 3
- Square root (√): The number that, when squared, gives the original
- Cube root (∛): The number that, when cubed, gives the original
Must-Know Facts
- Squares to 15²: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
- Cubes to 5³: 1, 8, 27, 64, 125
- (−3)² = 9 but −3² = −9 (brackets matter)
- √ symbol means positive root only (√9 = 3, not ±3)
- 2⁰ = 1; any non-zero number to the power 0 equals 1
- 2³ = 8 (not 6): powers mean multiply, not add
Key Formulas
- aⁿ = a × a × a × … (n times)
- √a = the number whose square is a
- ∛a = the number whose cube is a
- a^(1/2) = √a; a^(1/3) = ∛a (Higher tier)
- a^(−n) = 1/aⁿ (negative index = reciprocal)
Common Mistakes
- 2³ = 6: Wrong: 2³ = 2 × 2 × 2 = 8, not 2 × 3
- √(a + b) = √a + √b: False: √(9 + 16) = √25 = 5, NOT 3 + 4 = 7
- Negative index: 2⁻³ = −8 is wrong: 2⁻³ = 1/8 (reciprocal, not negative)
- Square root of a negative: √(−4) cannot be worked out, because no number multiplied by itself gives a negative answer
- Confusing √ with ÷ 2: √16 = 4 (not 8): root is NOT the same as halving
Worked Examples
Calculate: 3⁴ Solution 3⁴ means 3 × 3 × 3 × 3 = 9 × 3 × 3 = 27 × 3 = 81
Worked Examples
Find: √196 Solution Think: What number × itself = 196? Try 14: 14² = 196 ✓ Therefore √196 = 14 Memorising squares up to 20² helps!
Worked Examples
Calculate: (a) (-3)² (b) -3² Solution (a) (-3)² = (-3) × (-3) = 9 The brackets mean square the negative number (b) -3² = -(3 × 3) = -9 No brackets means square 3, then make negative Brackets make all the difference!
Worked Examples
Estimate √75 to 1 decimal place Solution √64 = 8 and √81 = 9 So √75 is between 8 and 9 75 - 64 = 11, 81 - 75 = 6 75 is closer to 81, so closer to 9 Try 8.7: 8.7² = 75.69 ≈ 76 Try 8.6: 8.6² = 73.96 ≈ 74 √75 ≈ 8.7