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When we talk about perfect squares, we refer to numbers that result from squaring a whole number. For example, when you square numbers like 1, 2, 3, and 4, you get perfect squares: 1, 4, 9, and 16, respectively. Recognizing these perfect squares is very useful, especially when simplifying radicals, because they help us to break down expressions more easily.
Identifying a perfect square can simplify the process significantly. For instance, if we have to simplify the expression \(\sqrt{245}\), finding that 49 is the largest perfect square that divides evenly into 245 is key. This recognition allows for the expression to be re-written, so we can extract the square root cleanly, which is a handy skill in algebra and pre-calculus. Understanding this concept will ultimately make simplifying radicals much easier.

Radical expressions involve roots, such as square roots, cube roots, etc. More often than not, they will appear with a square root symbol: \(\sqrt{}\). Learning to simplify radical expressions is fundamental in math, especially when you encounter them in equations or when solving for unknowns.
The expression \(2 \sqrt{245}\) in our original problem is a classic example of a radical expression. When simplifying, the goal is often to express the radical in its simplest form, which means finding if there are perfect square factors within the radicand (the number inside the radical). This is why understanding perfect squares, as mentioned above, plays a crucial role. By identifying the largest perfect square factor (which is 49) in 245, we can break down the radical into two parts: \(\sqrt{49}\) and \(\sqrt{5}\), simplifying the expression step by step.

The square root is one of the most common roots that you will encounter in mathematics. It asks the question: "What number multiplied by itself gives this number?" Simplifying a square root often involves recognizing whether the radicand is a perfect square or whether it can be broken down into a product involving a perfect square.
For instance, in \(\sqrt{49}\), determining that 49 is a perfect square, because it equals \(7^2\), lets us directly write the square root as 7. However, with non-perfect squares like 5 in \(\sqrt{5}\), the square root remains as is because 5 doesn't have an integer square root.
Combining these concepts, you can see how recognizing and using perfect squares, and knowing how to handle radical expressions and square roots, allows you to simplify expressions like \(2 \sqrt{245}\) to their simplest form, which in our solution simplifies to \(14 \sqrt{5}\).