What is a Square Root?
A square root of a number x is a value y such that y² = x. For example, √9 = 3 because 3² = 9. The symbol √ is called the radical sign. Every positive number has two square roots: one positive (principal square root) and one negative.
Types of Roots
- Square Root (√): The number that multiplied by itself gives the original number
- Cube Root (∛): The number that multiplied by itself three times gives the original
- Nth Root: General case where the result raised to power n equals the original
Perfect Squares
A perfect square is a number whose square root is an integer. The first 15 perfect squares are:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
Perfect Cubes
A perfect cube is a number whose cube root is an integer. The first 10 perfect cubes are:
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
Properties of Square Roots
- √(a × b) = √a × √b
- √(a / b) = √a / √b
- (√a)² = a
- √a² = |a| (absolute value)
Simplifying Radicals
A radical is in simplified form when the number under the radical sign has no perfect square factors (other than 1). For example, √12 = √(4×3) = 2√3.