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Imaginary numbers are an extension of the traditional number system. They come into play when we try to calculate the square root of negative numbers, which has no solution within the set of real numbers. In these scenarios, we introduce the unit imaginary number, denoted by \(i\), where \(i^2 = -1\). The introduction of \(i\) allows us to express roots of negative numbers as multiples of \(i\).
For example, the square root of \(-4\) can be written as \(2i\). This enables us to perform mathematical operations with numbers that have an imaginary component. In the current exercise, \(2i\) and \(-2i\) are examples of purely imaginary numbers, as there is no real part—only multiples of \(i\). These numbers can be raised to various powers just like real numbers, using the properties of \(i\).

Raising complex numbers to a power can initially seem daunting, but by understanding the properties of \(i\), it becomes straightforward. As we saw in the exercise, raising imaginary numbers like \(2i\) and \(-2i\) to the fourth power involves determining the power of \(i\) and the number separately.
Here's an essential property to remember:

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)

These powers of \(i\) repeat in a cycle every four powers. When raising a purely imaginary number, such as \(2i\), to the fourth power, it's a matter of applying these rules: \((2i)^4 = 2^4 \cdot i^4 = 16 \cdot 1 = 16\).
This method ensures that even powers of complex numbers simplify to a real number. The process is the same for negative purely imaginary numbers like \(-2i\), confirming their behavior in operations is regular and predictable. By systematically applying these rules, the seemingly complex calculations are easily managed.

Purely real numbers are those without any imaginary component. They exist along the real number line that we commonly use. In the exercise, the numbers \(2\) and \(-2\) are purely real numbers.
The operation of raising a real number to a power, such as the fourth power, is straightforward. There's no imaginary component to manage, so the rules of exponentiation we use are familiar and predictable.
For instance, \(2^4 = 16\) and \((-2)^4 = 16\), as any real number raised to an even power is positive. This particular property—where a negative number raised to an even power yields a positive result—occurs due to multiplying an even number of negative factors together, which pairs the negatives and results in a positive outcome. Thus in such exercises, purely real numbers behave in a consistency we are accustomed to in basic arithmetic.