Radical Expression
A radical expression is an algebraic expression that contains a radical symbol (√) indicating a root, such as a square root, cube root, or nth root of a number or polynomial. The number or expression under the radical is called the radicand, and the small number written in the notch of the radical symbol is the index indicating which root is taken. Simplifying radical expressions involves factoring out perfect powers and rationalizing denominators, which are essential skills in solving equations and computing exact values in trigonometry and geometry.
Key Formula
nth root of a = a^(1/n), nth root of a^m = a^(m/n)
LaTeX: \sqrt[n]{a} = a^{1/n},\quad \sqrt[n]{a^m} = a^{m/n}
| Symbol | Meaning | Unit |
|---|---|---|
| n | index (the degree of the root) | dimensionless |
| a | radicand (value under the radical) | dimensionless |
| m | exponent of the radicand | dimensionless |
Worked Example
Problem
Simplify √(72x³y²) assuming x, y ≥ 0.
Solution
Step 1: Factor the radicand: 72x³y² = 36·2·x²·x·y². Step 2: Group perfect squares: = (36·x²·y²)·(2x). Step 3: Apply product rule: √(36·x²·y²·2x) = √(36)·√(x²)·√(y²)·√(2x). Step 4: Simplify: = 6·x·y·√(2x). Step 5: Final simplified form: 6xy√(2x).
Answer
6xy√(2x)
Key Properties and Rules for Radical Expressions
| Property | Rule | Example | Result |
|---|---|---|---|
| Product rule | √(ab) = √a · √b | √(50) = √(25·2) | 5√2 |
| Quotient rule | √(a/b) = √a / √b | √(9/16) | 3/4 |
| Power rule | √(aⁿ) = a^(n/2) | √(x⁶) | x³ |
| Rationalise denom. | Multiply by conjugate | 1/√2 | √2/2 |
| Like radicals | Add/subtract same radicand | 3√5 + 2√5 | 5√5 |
| Nested radical | √(√a) = a^(1/4) | √(√81) | 3 |
Interactive Tools
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Related Terms
Polynomial
A polynomial is an algebraic expression consisting of variables and coefficients combined using addition, subtraction, and multiplication, where variables have non-negative integer exponents. The general form is aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where the highest exponent n is called the degree. Polynomials are used extensively in calculus, numerical analysis, and computer science for approximating functions and solving complex problems.
Quadratic Formula
The quadratic formula is an algebraic formula that gives the solutions (roots) of any quadratic equation ax² + bx + c = 0 directly in terms of its coefficients a, b, and c. It is derived by completing the square on the general quadratic and is the most reliable method for solving quadratics, working even when factoring over integers is impossible. The formula also reveals the nature of the roots through the discriminant b² − 4ac.
The word "radical" comes from the Latin "radix," meaning "root." The radical symbol (√) evolved from the letter "r" (for radix) used in medieval manuscripts and was modified over time. The modern form of the radical sign is attributed to René Descartes in the 17th century, who added the overline (vinculum) to indicate the extent of the radicand.