Rules of Square Roots

Direct Answer

The square root notation √x denotes the nonnegative number which, when squared, equals x. In symbols, √x = y means y² = x and y ≥ 0.

Informal Interpretation

Geometrically, √x represents the length of a side of a square whose area is x. It strips away any sign ambiguity so you always get a nonnegative result.

Fundamental Properties

• Principal root: √x ≥ 0 for all real x ≥ 0.
• Inverse of squaring: (√x)² = x for x ≥ 0.
• Absolute value relation: √(a²) = |a| for all real a.
• No simple linearity: √(a + b) ≠ √a + √b in general.

Algebraic Rules (for nonnegative inputs)

• Product rule: √(a·b) = √a·√b when a ≥ 0 and b ≥ 0.
• Quotient rule: √(a/b) = √a/√b when a ≥ 0 and b > 0.
• Power rule: √(xᵏ) = x^(k/2) for x ≥ 0 and any real k.

Domain Restrictions

• The expression √x is real only when x ≥ 0.
• For nested radicals, each radicand must satisfy its own nonnegativity condition.
• In equations, check both sides after squaring to avoid extraneous solutions.

Examples

1. √16 = 4 because 4² = 16.
2. √(9·25) = √9·√25 = 3·5 = 15.
3. √(49/64) = √49/√64 = 7/8.
4. Solve √(x + 3) = 5: square both sides to get x + 3 = 25, then x = 22.

Common Misconceptions

• People sometimes think √a + b means √a + √b, but it does not.
• After squaring an equation, negative roots can appear; always verify x ≥ –constant.

Next Steps

1. Explore nth roots notation: ⁿ√x = x^(1/n).
2. Study complex square roots where radicands can be negative.
3. Investigate radical equations and techniques for isolating and solving them.
4. Apply these rules in simplifying expressions and in solving geometry and calculus problems.