91Ó°ÊÓ

In this exercise, we are asked to find the product (11 - 2i)(11 + 2i). This is a great example of the algebraic concept known as the 'difference of squares'.
A difference of squares looks like: (a - b)(a + b) and simplifies to: a^2 - b^2.
Here, it's easy to see that:

• a = 11
• b = 2i

To apply the difference of squares, we recognize that: (11 - 2i)(11 + 2i) changes to:
11^2 - (2i)^2. This single step transforms our product into something easier to handle, so we can now move on to the calculation.

Complex numbers include a real part and an imaginary part, where 'i' represents the imaginary unit.
Multiplying complex numbers can be tricky, but understanding how the imaginary unit works makes it easier. In our example, 'i' stands for the square root of -1, meaning i^2 = -1.
When we apply this in our expression from the first section, we get:
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(2i)^2 = 4 * (i^2) = 4 * (-1) = -4

Rewriting our expression 11^2 - (2i)^2, we replace (2i)^2 with -4:
11^2 - (-4).
Thus, it is simplified: 121 + 4 = 125. Appreciating the role of the imaginary unit helps make complex number multiplication straightforward.

Understanding algebraic expressions is the foundation of solving problems like this one. A solid grasp of algebraic manipulation allows us to simplify and solve complex number problems.
Algebraic expressions combine numbers and variables using arithmetic operations. In our case, we use the expression (11 - 2i)(11 + 2i).
Steps to solve these algebraic problems involve recognizing patterns, such as the difference of squares, and then applying algebraic rules like expanding, factoring, and simplifying.
Takeaway: Recognizing common algebraic identities simplifies complex problems. Blending algebraic expressions with our understanding of complex numbers unlocks higher-level math solutions, making it possible to handle seemingly complicated tasks with ease.