91Ó°ÊÓ

Understanding the standard form of complex numbers is essential for performing operations like addition, subtraction, multiplication, and particularly, in this case, squaring. A complex number is typically expressed as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property that \(i^2 = -1\). Here, \(a\) is known as the real part and \(b\) the imaginary part of the complex number. This form is very helpful as it clearly differentiates the real and imaginary components, allowing for straightforward mathematical operations.

For example, in the exercise \( (2+3i)^2 + (2-3i)^2 \), both \(2+3i\) and \(2-3i\) are in standard form, which makes it easier to assess and apply algebraic rules.

Squaring a complex number is similar to squaring a binomial, but with one critical difference: the imaginary unit \(i\). When you square \(i\), you get \(i^2 = -1\). Hence, squaring the complex number \(a + bi\) uses the same formula \( (a + bi)^2 = a^2 + 2abi + (bi)^2 \), but you must remember to replace \(i^2\) with \( -1 \).

Let's investigate this with the provided exercise. To square \(2+3i\), we calculate \( 2^2 + 2 \cdot 2 \cdot 3i + (3i)^2 \), simplifying to \( 4 + 12i - 9 \) because \(i^2 = -1\). The same process is applied when squaring \(2-3i\), where we end up with \( 4 - 12i - 9 \) after simplification. Here, the middle term changes sign due to the \( -3i \) component in the original complex number.

The final step in dealing with complex numbers is often to simplify the expressions you've created through operations like squaring. To simplify, combine like terms and apply any necessary algebraic rules, such as the aforementioned \(i^2 = -1\). In our example, after squaring the complex numbers, you're left with two simpler expressions: for \( (2+3i)^2 \) you have \( -17 \) and for \( (2-3i)^2 \) you get \( 7 \).

Note that the imaginary parts canceled out because they were equal and opposite, which can often happen with conjugate pairs. Conjugates have the property that when squared and added, their imaginary parts cancel, leaving a real number as their sum, such as \( -10 \) in this case. This concept of simplifying is pivotal to solving complex expressions and getting to the final result in its simplest form.