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Imaginary numbers might sound like a work of a mathematical fantasy, but they are serious players in the world of complex numbers. Imaginary numbers involve the square root of negative one, designated as the imaginary unit 'i', where \( i^2 = -1 \). When a negative number appears under a radical sign, it's an indication that an imaginary number is present. For instance, in the exercise \( \sqrt{-7} \), the square root of -7 cannot be found among the real numbers, but in the imaginary number system it is expressed as \( \sqrt{7}i \).

This unique class of numbers extends the number system beyond the real numbers and is used in complex calculations that occur in fields such as engineering, physics, and even finance. Imaginary numbers let us solve equations that would otherwise have no solution and they interact under the same arithmetic rules as real numbers, with the additional twist that squaring them results in a negative product.

The binomial expansion comes into play when raising binomials to an exponent. A binomial is a simple algebraic expression that has two terms, such as \((a+b)\). The binomial theorem provides us with a formula to expand expressions like \((a+b)^n\) where 'n' is a whole number.

In the given exercise, we have the binomial \((-3-\sqrt{7}i)\) raised to the power of 2. The expansion follows the formula \((a - b)^2 = a^2 - 2ab + b^2\), with 'a' and 'b' as the two terms in the binomial. By applying this formula to our complex binomial, we avoid manually multiplying the binomial by itself, which can be more error-prone. The neat and predictable result of the binomial expansion helps us to efficiently handle expressions with both real and imaginary components in complex numbers.

The standard form of a complex number is written as \(a + bi\), where 'a' represents the real part, and 'bi' represents the imaginary part. This form is crucial for straightforwardly expressing and manipulating complex numbers.

With the exercise at hand, our goal is to simplify the expression until we can present it in this tidy format. After we perform the binomial expansion, we end up with real numbers and imaginary numbers mixed together. As the last step in solving the exercise, we separate these two types of numbers and combine like terms. This final simplification allows us to present the complex number in its standard form, which in this case is \(2 + 6\sqrt{7}i\). Representing complex numbers in standard form standardizes the process of doing arithmetic with them, such as addition, subtraction, and even multiplication or division with the help of additional rules.