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Square roots are a fundamental concept in algebra. They essentially "reverse" the operation of squaring a number. For example, the square root of 16 is 4, because when you square 4, you get 16. In radical equations like the one given, square roots can appear on one or both sides of the equation. Understanding how to isolate and remove the square root is key to solving such equations. In the equation \(\sqrt{3x - 2} = 4\), the square root is already isolated, which is the first step towards solving it. Once isolated, we can square both sides to remove the square root, simplifying the process of finding the solution.

Algebraic manipulation involves rearranging and solving equations to find the value of unknowns. When solving radical equations, estimating the involvement of square roots often requires the thoughtful use of algebraic manipulation.In the exercise, once the square root \(\sqrt{3x - 2}\) is squared, we obtain the equation \(3x - 2 = 16\). This is a simpler equation without radicals. From this point, algebraic manipulation involves:

• Moving terms to isolate the variable term, like adding 2 to both sides to get \(3x = 18\).
• Dividing by the coefficient of \(x\) to solve for the variable, resulting in \(x = 6\).

These steps highlight the method of isolating variables to determine their specific values effectively.

Checking solutions is a crucial step that ensures the accuracy of our solutions, particularly with radical equations. It's important because squaring both sides of an equation, which is frequently used to solve square root problems, can sometimes introduce extraneous or false solutions.To verify the solution \(x = 6\), we substitute it back into the original equation: \(\sqrt{3 \times 6 - 2} = 4\). Simplifying the expression under the square root gives \(\sqrt{16} = 4\). Since this equality holds true, \(x = 6\) is confirmed as the correct solution.By checking the solution, we confirm its validity and ensure that the answer satisfies the initial equation without any errors from previous steps.