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Simplifying expressions is a fundamental skill in algebra that involves reducing an expression to its simplest form. This often involves combining like terms, using properties of operations, and applying the rules of arithmetic. In the given exercise,
\[3 \sqrt{18}-\sqrt{8}\]
we focus on simplifying square roots, a specific case of simplification. It's essential to first identify the largest perfect square factor of the numbers inside the square root. A perfect square is a number that can be expressed as the product of an integer with itself. For example, 9 is a perfect square since it can be written as \(3^2\), and thus \(\sqrt{9} = 3\). This step reduces complex roots into more manageable components.

Understanding square root properties is critical when working with simplifying expressions involving roots. One key property we use is that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\), which allows us to simplify square roots of products. Also, it’s vital to recognize that the square root of a perfect square is simply the integer itself, thus \(\sqrt{a^2} = a\).
In our exercise, by recognizing that 18 is 9 times 2, and 8 is 4 times 2, we can simplify \(\sqrt{18}\) to \(\sqrt{9 \times 2}\) and \(\sqrt{8}\) to \(\sqrt{4 \times 2}\). Applying these properties efficiently is a stepping stone to mastering algebraic simplification.

Lastly, to fully simplify an expression, we must combine like terms. Like terms are terms within an algebraic expression that have the same variable raised to the same power.
In our example, after simplification, we end up with terms that have the same square root part: \(9\sqrt{2}\) and \(-2\sqrt{2}\). These are like terms because they share the \(\sqrt{2}\) component. To combine them, we simply add or subtract the coefficients (the numbers in front of the square roots), much like how you would combine similar items when making an inventory. This gives us \(7\sqrt{2}\), which is the simplest form of the original expression.