Evaluate Complex Expression: (8-3i)-(8-3i)(8+8i)

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Evaluate Complex Expression: (8-3i)-(8-3i)(8+8i)

Hey guys! Today, let's dive into a fun little math problem involving complex numbers. We're going to evaluate the expression: (8-3i) - (8-3i)(8+8i). Don't worry, it's not as scary as it looks! We'll break it down step by step, so you can follow along easily. Get your imaginary units ready; let's get started!

Understanding Complex Numbers

Before we jump into solving the expression, let's quickly recap what complex numbers are all about. A complex number is basically a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit. The imaginary unit i is defined as the square root of -1 (i.e., i² = -1). Complex numbers have a real part (a) and an imaginary part (bi). They're super useful in various fields like engineering, physics, and even computer graphics. In our expression, (8 - 3i) and (8 + 8i) are both complex numbers.

When dealing with complex numbers, we often need to perform operations like addition, subtraction, multiplication, and division. These operations follow specific rules to ensure we correctly handle the real and imaginary parts. For example, when adding or subtracting complex numbers, we combine the real parts together and the imaginary parts together. When multiplying complex numbers, we use the distributive property and remember that i² = -1. Understanding these basics is crucial for tackling more complex expressions, like the one we're about to evaluate.

Now, let's talk about why complex numbers are so important. They allow us to solve equations that have no real solutions. For instance, the equation x² + 1 = 0 has no real solutions because no real number squared can give you -1. But with complex numbers, we can say that x = ±i are the solutions. Complex numbers also provide a way to represent and manipulate two-dimensional quantities, which is why they're used extensively in fields dealing with vectors and oscillations. So, even though they might seem a bit abstract, complex numbers are incredibly practical and powerful tools.

Step-by-Step Evaluation

Okay, let's get back to our expression: (8-3i) - (8-3i)(8+8i). To evaluate this, we need to follow the order of operations (PEMDAS/BODMAS). That means we'll handle the multiplication first, and then the subtraction.

Step 1: Multiply the Complex Numbers

We need to multiply (8 - 3i) by (8 + 8i). We can use the distributive property (also known as the FOIL method) to do this:

(8 - 3i)(8 + 8i) = 8 * 8 + 8 * 8i - 3i * 8 - 3i * 8i

Let's break that down:

  • 8 * 8 = 64
  • 8 * 8i = 64i
  • -3i * 8 = -24i
  • -3i * 8i = -24i²

Now, remember that i² = -1, so -24i² = -24 * (-1) = 24. Putting it all together:

64 + 64i - 24i + 24 = (64 + 24) + (64i - 24i) = 88 + 40i

So, (8 - 3i)(8 + 8i) = 88 + 40i.

Step 2: Subtract the Result from (8 - 3i)

Now we have to subtract the result we just got from (8 - 3i):

(8 - 3i) - (88 + 40i) = 8 - 3i - 88 - 40i

Combine the real parts and the imaginary parts:

(8 - 88) + (-3i - 40i) = -80 - 43i

Therefore, (8-3i) - (8-3i)(8+8i) = -80 - 43i.

Detailed Breakdown of the Multiplication Step

Let's take a closer look at the multiplication step to ensure we've got it nailed down. We're multiplying (8 - 3i) by (8 + 8i). Here’s how it breaks down:

  1. First terms: Multiply the first terms in each complex number: 8 * 8 = 64. This gives us the first part of our result.
  2. Outer terms: Multiply the outer terms: 8 * 8i = 64i. This contributes to the imaginary part of our result.
  3. Inner terms: Multiply the inner terms: -3i * 8 = -24i. This also contributes to the imaginary part.
  4. Last terms: Multiply the last terms: -3i * 8i = -24i². Remember that i² = -1, so this becomes -24 * (-1) = 24. This contributes to the real part of our result.

Adding these all together, we get 64 + 64i - 24i + 24. Combining the real parts (64 + 24) gives us 88, and combining the imaginary parts (64i - 24i) gives us 40i. So, (8 - 3i)(8 + 8i) = 88 + 40i.

Understanding this step in detail can help clarify any confusion about how to multiply complex numbers. It’s all about carefully applying the distributive property and remembering the crucial fact that i² = -1. Once you’ve mastered this, you can confidently tackle more complex expressions involving complex numbers.

Common Mistakes to Avoid

When working with complex numbers, there are a few common mistakes that you should watch out for:

  1. Forgetting that i² = -1: This is probably the most common mistake. Always remember to replace i² with -1 when you see it.
  2. Incorrectly applying the distributive property: Make sure you multiply each term in the first complex number by each term in the second complex number.
  3. Mixing up real and imaginary parts: Keep the real and imaginary parts separate when adding or subtracting complex numbers.
  4. Not following the order of operations: Remember to perform multiplication before addition or subtraction.

By being aware of these potential pitfalls, you can avoid making mistakes and ensure that you get the correct answer every time.

Real-World Applications

Complex numbers might seem like an abstract concept, but they actually have many real-world applications. Here are a few examples:

  • Electrical Engineering: Complex numbers are used to analyze alternating current (AC) circuits. They can represent the magnitude and phase of voltage and current.
  • Quantum Mechanics: Complex numbers are fundamental to quantum mechanics, where they are used to describe wave functions and probabilities.
  • Signal Processing: Complex numbers are used in signal processing to analyze and manipulate signals. They are particularly useful for Fourier analysis.
  • Control Systems: Complex numbers are used to design and analyze control systems, which are used in everything from airplanes to robots.

So, the next time you're working with complex numbers, remember that you're not just doing abstract math – you're learning tools that are used in many important fields.

Conclusion

Alright, guys, we've successfully evaluated the expression (8-3i) - (8-3i)(8+8i)! We broke it down step by step, multiplied the complex numbers, and then subtracted the result. Remember, the final answer is -80 - 43i. Keep practicing with complex numbers, and you'll become a pro in no time. Keep up the great work, and I'll see you in the next math adventure!