Square Root Laws
Square Root Laws and Rules
These are the fundamental rules that govern square root arithmetic. Knowing them allows you to simplify radical expressions quickly.
1. Product Rule
√(a × b) = √a × √b
Example: √(4 × 9) = √4 × √9 = 2 × 3 = 6 | √36 = 6 ✓
2. Quotient Rule
√(a ÷ b) = √a ÷ √b
Example: √(100 ÷ 4) = √100 ÷ √4 = 10 ÷ 2 = 5 | √25 = 5 ✓
3. Power Rule
√(a²) = |a| and (√a)² = a
Example: √(7²) = √49 = 7 | (√5)² = 5
4. Addition Rule (Important!)
√a + √b ≠ √(a + b) ← This is WRONG
You can only add square roots if they have the same radicand (number under the root). Example: 3√2 + 5√2 = 8√2. But √2 + √3 cannot be simplified.
5. Negative Rule
√(−a) = i√a (where i = √−1)
Square roots of negative numbers are imaginary. Example: √−9 = 3i
6. Zero Rule
√0 = 0
Zero is the only number whose square root equals itself.