Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. √(2/3x)

Verified step by step guidance

1. Identify the expression given: \(\frac{\sqrt{2}}{3x}\). The numerator is the square root of 2, and the denominator is the product of 3 and \(x\).
2. Since the expression is a fraction with a square root in the numerator and a product in the denominator, check if any simplification is possible by factoring or rationalizing.
3. Note that the denominator \$3x$ does not contain any radicals, so rationalization is not necessary here.
4. Because \(x\) is a positive real number, the expression cannot be simplified further by combining terms under the square root or canceling factors.
5. Therefore, the simplified form of the expression remains \(\frac{\sqrt{2}}{3x}\).

Key Concepts

Simplifying Radicals

Simplifying radicals involves rewriting a square root expression in its simplest form by factoring out perfect squares. For example, √18 can be simplified to 3√2 because 18 = 9 × 2 and √9 = 3. This process makes expressions easier to work with and understand.

Properties of Square Roots

The square root of a product equals the product of the square roots: √(a × b) = √a × √b. This property allows us to separate or combine radicals, which is essential when simplifying expressions involving roots and variables.

Operations with Variables under Radicals

When variables are under a square root, their exponents affect simplification. For positive variables, √(x^2) = x because squaring and square rooting are inverse operations. Understanding this helps simplify expressions like √(3x) by treating coefficients and variables separately.