What is √(28x²)?

The expression √(28x²) simplifies to 2x√7 when x is positive, or more generally, 2|x|√7 when x can be any real number. This problem combines the simplification of numerical radicals with the handling of variables under square roots, a skill that's essential in Class 10 algebra. Understanding how to simplify radical expressions involving variables prepares you for solving quadratic equations, working with polynomial expressions, and tackling more advanced algebraic manipulations.

Let's break down the simplification process step by step. Starting with √(28x²), we can use the property that the square root of a product equals the product of the square roots: √(28x²) = √28 × √(x²). We've already established that √28 = 2√7. For √(x²), the result depends on whether x is positive or negative. If x is positive (x ≥ 0), then √(x²) = x. If x could be negative, then √(x²) = |x| (the absolute value of x), because the square root symbol always represents the non-negative result. For example, if x = -3, then x² = 9, and √9 = 3 = |x|, not -3. Therefore, the complete simplification is: √(28x²) = √28 × √(x²) = 2√7 × |x| = 2|x|√7. However, in many Class 10 problems, when variables appear in radical expressions, we assume they represent positive values unless otherwise stated, so the answer is often written simply as 2x√7.

The simplification can also be approached by factoring everything inside the radical first: √(28x²) = √(4 × 7 × x²) = √4 × √7 × √(x²) = 2 × √7 × x = 2x√7 (assuming x ≥ 0). This method emphasizes the systematic extraction of perfect square factors—both numerical (4 = 2²) and algebraic (x²). The key insight is that any even power of a variable is a perfect square: x² = (x)², x⁴ = (x²)², x⁶ = (x³)², and so on. Therefore, √(x²) = x, √(x⁴) = x², √(x⁶) = x³, and this pattern continues.

Understanding how to work with variables under radicals has important applications in various mathematical contexts. In geometry, when using the Pythagorean theorem with algebraic expressions, you might encounter expressions like √(9x² + 16x²) = √(25x²) = 5x. In solving quadratic equations using the quadratic formula, you work with expressions like √(b² - 4ac), which might simplify when substituted with specific values. In graphing radical functions, simplifying expressions like y = √(4x²) to y = 2|x| helps you understand the shape and behavior of the function.