Question

Three radical expressions have different radicands and, when simplified, are like radicals to squareroot 3xy.Describe key characteristics of these radical expressions.

Answer 1

Key fact: a square root simplifies to a multiple of √(3xy) exactly when its radicand equals 3xy times a perfect square.

Characteristics
- Each original radicand has the form n^2·3xy for some positive integer n (or for some perfect-square factor n^2).
- After simplifying √(n^2·3xy) = n·√(3xy), so all simplified radicals are "like" (same radical part √(3xy)) and differ only by the numeric coefficient n.
- The radicands can be different because they use different perfect squares n^2 (for example 1, 4, 9, 16, …).
- For real principal square roots you must have 3xy ≥ 0 (commonly one assumes x ≥ 0 and y ≥ 0).

Examples
- √(3xy) = 1·√(3xy) (here n = 1, radicand = 1·3xy)
- √(12xy) = √(4·3xy) = 2·√(3xy) (n = 2, radicand = 4·3xy)
- √(27xy) = √(9·3xy) = 3·√(3xy) (n = 3, radicand = 9·3xy)

General form: any radical of the form √(n^2·3xy) will simplify to n√(3xy).