Simplify Complex Number Expression: (6-4i)² - 3(7-11i)

Hey math whizzes and number crunchers! Today, we're diving deep into the fascinating world of complex numbers. Get ready to roll up your sleeves and simplify an expression that looks a bit intimidating at first glance: $(6-4i)(6-4i)-3(7-11i)$. Our mission, should we choose to accept it, is to transform this beast into the standard form of a complex number, $a+bi$, where $a$ and $b$ are nice, clean rational numbers.

Understanding the Basics: What's an Imaginary Number?

Before we jump into the nitty-gritty, let's do a quick refresh on what we're dealing with. You've probably seen numbers like 1, 2, 3.5, or -1/2. These are our good ol' real numbers. But then, things get a little more interesting with imaginary numbers. The star of the show here is 'i'. You'll remember that $i$ is defined as the square root of -1 (that is, $i = \sqrt{-1}$). This little guy unlocks a whole new realm of numbers, the complex numbers, which are crucial in many areas of science, engineering, and even art. A complex number is typically written in the form $a+bi$, where $a$ is the real part and $b$ is the imaginary part. Our goal is to isolate these real and imaginary parts for the given expression.

Step 1: Tackling the Squaring Part - $(6-4i)(6-4i)$

Alright guys, let's start with the first part of our expression: $(6-4i)(6-4i)$. This is essentially $(6-4i)^2$. To handle this, we can use the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last). Let's break it down:

• First: Multiply the first terms in each binomial: $6 * 6 = 36$.
• Outer: Multiply the outer terms: $6 * (-4i) = -24i$.
• Inner: Multiply the inner terms: $(-4i) * 6 = -24i$.
• Last: Multiply the last terms: $(-4i) * (-4i) = 16i^2$.

Now, let's put it all together: $36 - 24i - 24i + 16i^2$.

Remember that crucial property of $i$: $i^2 = -1$. So, we can substitute $-1$ for $i^2$ in our expression: $16i^2 = 16*(-1) = -16$.

Now, combine the real parts and the imaginary parts:

• Real parts: $36 - 16 = 20$.
• Imaginary parts: $-24i - 24i = -48i$.

So, $(6-4i)^2$ simplifies to $20 - 48i$. Pretty neat, huh?

Step 2: Handling the Second Part - $-3(7-11i)$

Next up, let's simplify the second part of our original expression: $-3(7-11i)$. This is a straightforward distributive property application. We need to multiply $-3$ by each term inside the parentheses:

• Multiply $-3$ by $7$: $-3 * 7 = -21$.
• Multiply $-3$ by $-11i$: $-3 * (-11i) = 33i$.

So, $-3(7-11i)$ simplifies to $-21 + 33i$. Easy peasy!

Step 3: Combining the Simplified Parts

Now for the grand finale! We've simplified both parts of our original expression. We have $20 - 48i$ from the first part and $-21 + 33i$ from the second part. Our original expression was $(6-4i)^2 - 3(7-11i)$. So, we need to combine these two results:

$(20 - 48i) + (-21 + 33i)$

To combine complex numbers, we simply add the real parts together and add the imaginary parts together:

• Combine the real parts: $20 + (-21) = 20 - 21 = -1$.
• Combine the imaginary parts: $-48i + 33i = (-48 + 33)i = -15i$.

Putting it all together, our final simplified expression is $-1 - 15i$.

Final Answer: The Form $a+bi$

We've successfully transformed the original expression into the form $a+bi$. In our case, $a = -1$ and $b = -15$. Both $-1$ and $-15$ are rational numbers, which is exactly what the question asked for. So, the expression $(6-4i)(6-4i)-3(7-11i)$ written in the form $a+bi$ is $-1 - 15i$.

Why This Matters: The Power of Complex Numbers

So, why do we bother with these complex numbers, you ask? Well, they're not just a mathematical curiosity, guys! Complex numbers are absolutely fundamental in many scientific and engineering fields. Think about electrical engineering – they use complex numbers to analyze alternating current (AC) circuits. In quantum mechanics, they're essential for describing wave functions. Signal processing, control theory, fluid dynamics – you name it, and complex numbers probably play a role. Being able to manipulate and simplify expressions involving complex numbers, like we just did, is a foundational skill that opens doors to understanding more advanced concepts. It's like learning your ABCs before you can read a novel; mastering these basics allows you to tackle more complex problems down the line. Keep practicing, and you'll be a complex number pro in no time!

Key Takeaways for Your Math Toolkit

• Definition of $i$: Remember that $i = \sqrt{-1}$ and $i^2 = -1$. This is the golden rule for simplifying complex expressions.
• FOIL Method: Use FOIL (or the distributive property) when multiplying binomials involving complex numbers.
• Combining Like Terms: Always group and combine the real parts and the imaginary parts separately.
• Standard Form $a+bi$: Aim to express your final answer in this format, clearly identifying the real part ($a$) and the imaginary part ($b$).

Mastering these steps will make simplifying complex number expressions a breeze. So next time you see an expression like this, don't sweat it! Just break it down, apply the rules, and you'll arrive at the answer with confidence. Keep exploring the amazing world of mathematics, and remember, practice makes perfect!