Power and Roots (Copy)

1. Squares and Square Roots (² and √ )

• Square of a number: Multiply the number by itself.
  n² = n × n
  Example: 7² = 7 × 7 = 49
• Square Root (√n): A number which, when multiplied by itself, gives n.
  Example: √49 = 7
• Key Squares (1² to 15²)

• Examples:
  • Write down √169 → 13
  • Calculate 12² → 144

2. Cubes and Cube Roots (³ and ∛ )

• Cube of a number: Multiply the number by itself twice.
  n³ = n × n × n
  Example: 3³ = 3 × 3 × 3 = 27
• Cube Root (∛n): A number which, when cubed, gives n.
  Example: ∛27 = 3
• Key Cubes and Cube Roots

• Examples:
  • Find ∛125 → 5
  • Calculate 4³ → 64

3. Other Powers (n^x) and Roots

• General Powers:
  n^x means multiplying n by itself x times.
  Example: 2⁴ = 2 × 2 × 2 × 2 = 16
• Common Powers:
  • n¹ = n
  • n² = Square
  • n³ = Cube
  • n⁰ = 1 (any number to the power 0 is 1)
  • n^½ = √n (square root)
  • n^(1/3) = ∛n (cube root)
• Roots:
  • Fourth Root: n^(1/4)
  • Fifth Root: n^(1/5)
    Example: 16^(1/4) = 2

4. Key Rules of Powers

• Multiplying powers (same base):
  a^m × a^n = a^(m+n)
  Example: 2³ × 2² = 2⁵ = 32
• Dividing powers (same base):
  a^m ÷ a^n = a^(m−n)
  Example: 5⁴ ÷ 5² = 5² = 25
• Power of a power:
  (a^m)^n = a^(m×n)
  Example: (3²)³ = 3⁶ = 729
• Negative powers:
  a^(-n) = 1 / a^n
  Example: 2^(-3) = 1 / 8

5. Example Calculations

• Write down the value of √169
  Answer: 13
• Calculate 5 × 8² ÷ 3
  Step 1: 8² = 64
  Step 2: 5 × 64 = 320
  Step 3: 320 ÷ 3 = 106.67
• Simplify (2³ × 2²)
  2³ × 2² = 2^(3+2) = 2⁵ = 32
• Evaluate (4²)³
  4² = 16
  16³ = 16 × 16 × 16 = 4096
• Find the value of 1000^(1/3)
  ∛1000 = 10

6. Quick Reference Table

7. Tips for Exams

• Memorize squares from 1² to 15² and cubes of 1, 2, 3, 4, 5, 10.
• Recognize that:
  • √ means square root.
  • ∛ means cube root.
• Watch out for BIDMAS (Brackets, Indices, Division/Multiplication, Addition/Subtraction) when calculating.
• Negative and fractional powers often appear in trick questions — remember their rules.
• For non-calculator papers, quick recall saves time—especially for squares and cubes.