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In any complex number, like the ones seen in the exercise, there are two distinct components: the real part and the imaginary part. The real part is basically the component that doesn't involve the imaginary unit \(i\), which satisfies the equation \(i^2 = -1\). The real part is simply a standard number from the set of real numbers.

For example, in the complex number \(7 + 2i\), the real part is 7. In the second number, \(1 - 4i\), the real part is 1. By identifying these parts, you can easily focus on combining them during operations like addition or subtraction. This fundamental skill helps in simplifying the process of working with complex numbers.

The imaginary part of a complex number involves the imaginary unit \(i\). Imaginary numbers are those which give a negative product when squared; specifically, \(i^2 = -1\). The imaginary part of a complex number is usually written right after the real part and involves \(i\).

In the exercise, look at the complex numbers \(7 + 2i\) and \(1 - 4i\). We can easily identify their imaginary parts: 2i and \(-4i\) respectively.

When performing operations on complex numbers, you add or subtract the imaginary parts separately from the real parts. This means, for our exercise, the imaginary parts \(2i\) and \(-4i\) are combined to form \(-2i\). Understanding these imaginary components makes complex numbers much easier to manage.

The concept of standard form is essential in understanding complex numbers. Complex numbers are typically written in the standard form as \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part, with \(b\) being a real number and \(i\) the imaginary unit.

In our example, after performing the required operations, combining the real and imaginary parts results in \(8 - 2i\). This is the result expressed in standard form. Having complex numbers in this form allows for consistency and makes further calculations and comparisons straightforward. Therefore, writing complex numbers in standard form is a critical step to ensure clarity and precision in mathematical solutions involving complex numbers.