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The process of simplifying expressions is crucial in mathematics to make complex problems more manageable. Simplification involves reducing an expression to its simplest form while still maintaining its original value. In the given exercise, simplifying the expression starts with combining like terms under the square root, specifically adding the numbers 144 and 25 together to get 169. This is a crucial step as it transforms the initial expression into a simpler form, \(\sqrt{169}\), which can then be easily evaluated.

When simplifying, one should always look for opportunities to combine, factor, or cancel terms to reduce the complexity of the expression. Remember that the goal is not to change the value of the expression but to make it as straightforward as possible for evaluation or further manipulation.

Radical expressions involve roots, such as square roots, cube roots, and so on. In our exercise, \(\sqrt{144 + 25}\) is a radical expression, with the radical sign \(\sqrt{}\) denoting the square root. To evaluate such expressions, we begin by simplifying the number inside the radical sign, if possible. In this case, simplifying the expression under the square root allows us to rewrite it as \(\sqrt{169}\).

Understanding that square roots are the opposite of squaring a number is essential. When we take the square root of a number, we're looking for the value that, when squared, would result in the original number inside the radical. For the expression \(\sqrt{169}\), we seek a number which, when multiplied by itself (squared), gives us 169. That number happens to be 13, which is why the solution to the expression is 13.

Real numbers form a continuum that includes rational numbers, such as integers and fractions, and irrational numbers, such as the square root of 2 (\(\sqrt{2}\)). When evaluating square roots, the result is not always a neat integer or fraction. However, the exercise given guarantees a real number because both 144 and 25 are perfect squares, and their sum, 169, is also a perfect square, which means its square root is an integer.

The set of real numbers is closed under addition and square roots of non-negative numbers, meaning that adding real numbers together or taking a square root of a non-negative real number will always yield another real number. Therefore, if the expression under the square root is a positive real number, as in this exercise, the square root is also a real number, specifically 13 in this example.