3sqrt2

3√2 denotes the real number obtained by multiplying 3 by the square root of 2. It can also be written as √18, since √2 × 3 = √(2 × 9) = √18. Numerically, 3√2 is approximately 4.242640687, and is often rounded to 4.24264 in quick calculations.

As a radical, 3√2 is an irrational number; it cannot be expressed as a ratio of integers. It is algebraic of degree 2 over the rational numbers, because it is a root of the polynomial x^2 − 18 = 0. Consequently it belongs to the quadratic field Q(√2). The value remains unchanged under addition or subtraction of rational multiples, and it adheres to standard radical multiplication rules such as √a · √b = √(ab) for nonnegative a and b.

In geometry and applications, 3√2 commonly appears as the length of a diagonal in a square of side length 3, obtained via the Pythagorean theorem: d = √(3^2 + 3^2) = √18 = 3√2. It also arises in distance calculations between points like (0,0) and (3,3), where the Euclidean distance is √18. Beyond simple geometry, 3√2 serves as a representative example of irrational radical expressions used in algebra, analysis, and measurement contexts.