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When we delve into the realm of complex numbers, we encounter the 'imaginary unit,' denoted as 'i'. This unit is defined as the square root of -1. It may sound abstract since no real number multiplied by itself gives a negative product. However, in the context of complex numbers, this 'imaginary unit' is crucial for expressing numbers that cannot be represented on the real number line.

For example, consider the square root of -75. It involves a negative number under the square root, which is where the imaginary unit comes into play. By separating the negative sign and rewriting it as \(i\sqrt{75}\), we can work with it like any other algebraic expression. This separation is possible due to the property that \(\sqrt{a*b} = \sqrt{a}\sqrt{b}\), where \(a\) and \(b\) are any real numbers. It allows us to simplify complex square roots using the imaginary unit 'i' as a substitute for \(\sqrt{-1}\).

Simplifying square roots is an essential skill when working with both real and complex numbers. To simplify a square root, we look for factors within the radical that are perfect squares. For the given exercise, the number under the square root is 75. To simplify \(\sqrt{75}\), we identify that 25 is a perfect square factor of 75 (since 25 * 3 = 75). We can then rewrite the square root as \(\sqrt{25*3}\), which simplifies to \(5\sqrt{3}\). This way, we break down a complex square root into a simpler form that's more manageable to work with.

Following these steps not only makes the number easier to understand but also prepares it for further operations, such as addition, subtraction, or, as in our exercise, squaring the number. Breaking down square roots into their simplest form is a universally applicable technique that helps in various mathematical contexts.

Squaring complex numbers involves dealing with both their real and imaginary components. A complex number is squared by multiplying it by itself. In our example, we have \(5i\sqrt{3}\) which needs to be squared. Let's remember that \(i^{2} = -1\) whenever we square the imaginary unit. Therefore, squaring \(5i\sqrt{3}\) gives \(25i^{2}*3\), and since \(i^{2} = -1\), it simplifies to \(25 * -1 * 3\), resulting in -75.

It's a straightforward process once you remember the rules for squaring 'i' and apply the multiplication property consistently. When squaring complex numbers, it's essential to handle each part—the real and the imaginary—carefully and use their properties to simplify the expression effectively. This understanding paves the way for mastering more complex operations in the vast field of complex numbers.