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The distributive property is a fundamental principle in algebra that allows us to multiply a single term by each term within a parenthesis. When dealing with complex numbers, like in the exercise \(1+i)(3-2i)\), we apply the distributive property by multiplying each term in the first complex number by each term in the second complex number.

Thus, you have two products to calculate: the first number, 1, multiplied by both terms in the second number, \(3-2i\), giving us \(3-2i\), and the second term, \(i\), multiplied by both terms in the second number, resulting in \(3i-2i^2\). This property is essential as it consistently leads to the correct expansion of expressions involving any kind of numbers, whether they are real, complex, or even polynomials.

Complex numbers consist of two parts: a real part and an imaginary part. The standard form of a complex number is expressed as \(a + bi\), where \(a\) represents the real part and \(b\) represents the imaginary part with \(i\) being the imaginary unit \(\sqrt{-1}\).

After applying the distributive property to complex numbers, the result must often be translated back into this standard form. This involves combining like terms and simplifying any powers of \(i\), such as \(i^2\), into their real number equivalents. The exercise provided is a perfect example of this, where we must manipulate the expanded product to isolate the real components \(a\) and the imaginary components \(b\) to achieve the standard form, \(a + bi\).

When simplifying expressions with complex numbers, 'combining like terms' is crucial. Like terms are terms in an expression that have the same variable parts, such as real numbers or multiples of \(i\). For instance, in our problem \(3-2i + 3i-2i^2\), we group the real parts (\(3\) and \(2\), from \(3 + 2\), after recognizing that \(i^2 = -1\)) and the imaginary parts (\(3i\) and \( -2i\), from \(3i - 2i\)).

The next step is to combine these like terms. Combining \(3\) and \(2\), we get \(5\). Combining \(3i\) and \( -2i\), we get \(i\). Putting these together gives us a simplified expression of \(5+i\), which is in the standard form for complex numbers. This step is essential for simplifying complex expressions into a form that's easy to understand and use for further calculations.