What is √(28x² ÷ 9)?

The expression √(28x² ÷ 9) can be simplified to (2x√7)/3 when x is positive, or more precisely, (2|x|√7)/3 for any real value of x. This problem combines several algebraic skills: simplifying radicals, working with variables, and handling division within radicals—all important competencies for Class 10 mathematics. Let's explore the complete simplification process and the mathematical principles involved.

Starting with √(28x² ÷ 9), we first rewrite division as a fraction: √(28x²/9). Using the property that √(a/b) = √a/√b, we can separate this into: √(28x²)/√9. For the numerator, √(28x²) = √28 × √(x²) = 2√7 × x = 2x√7 (assuming x ≥ 0). For the denominator, √9 = 3. Therefore: √(28x²/9) = (2x√7)/3. This is the simplified form, with the numerical coefficient (2/3) separated from the radical part (√7) and the variable (x).

Alternatively, we could approach this by separating all factors first: √(28x²/9) = √(4 × 7 × x² / 9) = (√4 × √7 × √x²) / √9 = (2 × √7 × x) / 3 = (2x√7)/3. Both methods yield the same result, demonstrating the flexibility of radical manipulation. The expression is fully simplified because: (1) all perfect square factors have been extracted from under the radical (4 from the numerator, 9 from the denominator), (2) the denominator contains no radicals (it's the rational number 3), and (3) the radical √7 cannot be simplified further since 7 is prime.

An important consideration when working with variables under radicals is the domain restriction. Strictly speaking, √(x²) = |x| (absolute value), not simply x, because the principal square root is always non-negative. However, in many Class 10 algebra problems, variables are assumed to represent positive values unless otherwise stated. If we maintain mathematical rigor, the complete answer would be (2|x|√7)/3, which equals (2x√7)/3 when x ≥ 0 and (-2x√7)/3 when x < 0. In practice, exam problems usually either explicitly state that variables are positive or the context makes this clear.

We can verify our simplification by squaring the result: [(2x√7)/3]² = (2x√7)² / 3² = (4x² × 7) / 9 = 28x²/9, which matches the original expression inside the radical, confirming our answer is correct. This verification technique—squaring your simplified radical expression to check if it equals the original expression—is a valuable habit that catches errors and builds confidence.