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When we talk about square roots, we refer to a number that, when multiplied by itself, equals the original number. The square root operation is denoted by the radical symbol \( \sqrt{\cdot} \). For example, the square root of 9 is 3, because \(3 \times 3 = 9\). Square roots can be either positive or negative, but by convention, we usually refer to the positive root unless otherwise specified.

Square roots apply directly to positive numbers and zero. However, taking the square root of a negative number is a different story, which we'll discuss more in the next section. Remember, the aim of the square root operation is to determine which number, when squared, results in the number under the radical. This means if the number inside the square root is negative, it becomes problematic in the realm of real numbers.

Negative numbers are values less than zero, appearing on the left side of the number line. They have unique properties when combined with other math concepts, especially square roots. Multiplying a negative number by itself results in a positive number, but square roots don't play nicely with negatives in the realm of real numbers.

No real number, when squared, results in a negative number; thus, square roots of negative numbers are not considered real. In the exercise, \(25 - 144 = -119\). Trying to evaluate \( \sqrt{-119} \) doesn’t yield a real number because there isn’t a real number that multiplied by itself results in \(-119\). Therefore, expressions like \( \sqrt{-119} \) fall outside the real number system. In more advanced math, such numbers are considered complex, but that's not within the scope of real numbers.

Evaluating an expression means calculating its value. For the expression \( \sqrt{25 - 144} \), this process involves simplifying and checking the values as you go.

• Simplify the Expression: Start by evaluating the arithmetic inside the parentheses or radicals. Here, compute \(25 - 144 = -119\).
• Analyze the Result: Check if the resulting number is something you can take the square root of within real numbers. Since \(-119\) is negative, you can't take a square root and get a real number.
• Determine the Conclusion: Conclude that when a radical results in a negative number, it doesn't produce a real number output.

Understanding these steps is crucial, as they help clarify whether certain mathematical operations yield real number results or not. This process of evaluation underscores whether a solution is possible within the constraints of real numbers.