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When it comes to understanding square roots of negative numbers, we're delving into the realm of complex numbers. Complex numbers are a type of number that include a real part and an imaginary part. The imaginary unit, denoted as 'i', is defined as the square root of -1, which means that \(i^2 = -1\). Any negative number can be expressed as a complex number by taking the square root of the corresponding positive number and multiplying it by 'i'. For example, the square root of -9 is \(3i\), because \(3^2 = 9\), and when we insert the negative sign, it becomes \(3i\) since \(i\) represents the square root of -1.

This might seem a little abstract at first, but it's a crucial concept that allows us to perform arithmetic with square roots of negative numbers as if they were any other numbers, opening up a whole new world of possibilities in mathematics and engineering!

Simplifying square roots, especially when they involve negative numbers, is an extension of dealing with complex numbers. To simplify a square root of a negative number, we first need to recognize that the result is going to be an imaginary number. As demonstrated in our problem, we converted \(\sqrt{-9}\), \(\sqrt{-16}\), and \(\sqrt{-25}\) into their equivalent imaginary forms, which were \(3i\), \(4i\), and \(5i\) respectively.

The key here is to remember that the square root function separates the negative sign out as 'i', and then we're left with finding the square root of a positive number, which we handle just like any other square root. When solving exercises involving square roots of negative numbers, identifying this step correctly is essential for achieving the correct answer.

The concept of 'like terms' deals with combining terms that have the same variables raised to the same power. In our exercise, combining like terms is the final step to simplifying the expression. Once we've converted all the square roots of negative numbers into their complex form, we look for terms that can be combined because they're 'like terms'.

In this scenario, all the terms contain the imaginary unit 'i', and since they're all raised to the first power, they can be merged. We then add or subtract the coefficients just as we would with regular numbers. This is how we reached the result \(3i + 4i - 5i - 5i - 5i = -8i\) in the given problem. Understanding like terms and how to combine them is vital for simplification of expressions in algebra and plays a major role in simplifying complex number expressions.