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Understanding how to simplify square roots is an essential skill in mathematics that can make solving problems much quicker and easier. The general process begins with breaking down the number inside the square root, known as the radicand, into its prime factors.

Prime factorization helps to find pairs of identical factors, since the square root of a number is essentially finding a value that, when multiplied by itself, gives the radicand. For any pair of identical factors, one factor can come out of the square root symbol as it represents the value multiplied by itself.

Take the exercise \(\sqrt{25-16}\), for instance. Before we can simplify the square root, we need to work out the arithmetic inside it. The expression inside simplifies to \(\sqrt{9}\), since 25 minus 16 equals 9. Nine is a perfect square, being \(3 \times 3\), so the square root of nine is three. This simplification process turns a radical expression into a simpler or even non-radical form.

The real numbers encompass a vast range of values, including integers, decimals, and fractions that we use regularly in calculations. They can be positive, negative, or zero. Importantly, we can plot every real number on a number line.

When evaluating a square root such as \(\sqrt{25-16}\), we're considering if the number under the square root is a non-negative real number because the square root of a negative number is not a real number; it is an imaginary number (unless we're working with complex numbers, which is a topic for another discussion).

Since 25-16 equals 9, and 9 is a positive real number, we can find its square root in the set of real numbers. Thus, the square root of 9 is 3, which is also a real number. If the result under the square root had been negative, it would not have been a real number, and we could not represent it on the real number line.

Radical expressions contain roots, such as square roots, and they are a critical part of algebra. The term 'radical' refers to the root symbol itself. For example, \(\sqrt{9}\) is a radical expression where 9 is the radicand. It's important to know that radical expressions can be both simplified and operated upon, much like other algebraic expressions.

However, to simplify a radical expression, you have to consider a few factors. If the radicand is a perfect square, like in our original exercise \(\sqrt{25-16}\), simplifying is straightforward because the square root of a perfect square is an integer. Otherwise, we may have to use other methods like prime factorization to simplify the radical as much as possible. The goal is to have the radicand not have any perfect square factors, apart from one, which we can then easily calculate the root of.