Solving Complex Numbers √(-72) + √(-8) In A + Bi Form

Hey guys! Today, we're diving into the fascinating world of complex numbers and tackling a problem that might seem a bit tricky at first glance. We're going to perform the indicated operations and express our final answer in the standard form of a complex number: a + bi. Don't worry; it's not as daunting as it sounds! We'll break it down step by step so you can master these types of problems.

Breaking Down Complex Numbers

Before we jump into the problem, let's refresh our understanding of complex numbers. At its core, a complex number is a combination of a real number and an imaginary number. You're probably familiar with real numbers – they're the numbers we use every day, like 1, -5, 3.14, and so on. Imaginary numbers, on the other hand, involve the square root of -1, which is denoted by the symbol i. This 'i' is the key to unlocking the world of complex numbers.

So, a complex number is generally written in the form a + bi, where a represents the real part, and bi represents the imaginary part. The real part can be any real number, and the coefficient b in the imaginary part is also a real number. Think of it as a recipe: you need both the real and imaginary ingredients to bake a complex number!

Now, why do we even need complex numbers? Well, they pop up in various areas of mathematics, physics, and engineering. They're particularly useful for solving equations that don't have real number solutions, dealing with alternating current circuits, and even in quantum mechanics. So, understanding complex numbers is crucial for anyone venturing into these fields.

Tackling the Problem: √(-72) + √(-8)

Alright, let's get our hands dirty with the problem at hand: √(-72) + √(-8). The first thing we notice is that we're dealing with the square roots of negative numbers. This is where the imaginary unit i comes to our rescue. Remember, i is defined as the square root of -1 (i = √(-1)). This is a fundamental concept when working with square roots of negative numbers.

Our mission is to simplify these square roots and express them in terms of i. To do this, we'll use a little trick: we'll rewrite the negative numbers inside the square roots as products involving -1. For example, -72 can be written as -1 * 72, and -8 can be written as -1 * 8. This might seem like a simple step, but it's crucial for isolating the imaginary unit.

Let's apply this to our problem:

√(-72) + √(-8) = √(−1 * 72) + √(−1 * 8)

Now, we can use the property of square roots that states √(a * b) = √a * √b. This allows us to split the square roots:

√(−1 * 72) + √(−1 * 8) = √(-1) * √72 + √(-1) * √8

Remember that √(-1) is just i, so we can substitute that in:

√(-1) * √72 + √(-1) * √8 = i * √72 + i * √8

Great! We've successfully introduced the imaginary unit i into our expression. But we're not done yet. We need to simplify the square roots of 72 and 8. This is where our knowledge of perfect squares comes in handy.

Simplifying the Square Roots

To simplify √72, we need to find the largest perfect square that divides 72. A perfect square is a number that can be obtained by squaring an integer (e.g., 4, 9, 16, 25, etc.). In this case, the largest perfect square that divides 72 is 36 (since 36 * 2 = 72). We can rewrite √72 as follows:

√72 = √(36 * 2) = √36 * √2 = 6√2

Similarly, for √8, the largest perfect square that divides 8 is 4 (since 4 * 2 = 8). So, we can simplify √8 as:

√8 = √(4 * 2) = √4 * √2 = 2√2

Now, let's substitute these simplified square roots back into our expression:

i * √72 + i * √8 = i * (6√2) + i * (2√2)

Combining Like Terms

We're almost there! We now have an expression with two terms, both involving i and √2. These are like terms, which means we can combine them just like we combine regular algebraic terms.

Think of i√2 as a single entity, like 'x' in an algebraic expression. We have 6 of these entities plus 2 of these entities, which gives us a total of 8 of these entities. So, we can write:

i * (6√2) + i * (2√2) = 6i√2 + 2i√2 = (6 + 2)i√2 = 8i√2

The Final Answer: Expressing in a + bi Form

We've simplified our expression as much as possible. We're left with 8i√2. Now, we need to express this in the standard form of a complex number, a + bi. In this case, our real part a is 0 (since there's no real number term), and our imaginary part b is 8√2. Therefore, the final answer in a + bi form is:

0 + 8√2i

And that's it! We've successfully performed the indicated operations and expressed the answer in the required form. You might want to use a complex number calculator to make your work easier in the future. But doing it the long way helps a lot to master the concepts.

Key Takeaways and Tips

Let's recap the key steps we took to solve this problem:

1. Recognize the imaginary unit: Remember that i = √(-1). This is the foundation for working with square roots of negative numbers.
2. Rewrite negative square roots: Express the negative number inside the square root as a product involving -1. This allows you to isolate the imaginary unit.
3. Simplify square roots: Find the largest perfect square that divides the number inside the square root and simplify accordingly.
4. Combine like terms: If you have terms with the same imaginary unit and radical, you can combine them just like you combine algebraic terms.
5. Express in a + bi form: Make sure your final answer is in the standard form of a complex number, a + bi, where a is the real part, and bi is the imaginary part.

Here are a few extra tips to help you conquer complex number problems:

• Practice makes perfect: The more you practice, the more comfortable you'll become with complex numbers. Try solving various problems with different levels of difficulty. Practice complex number calculations regularly.
• Pay attention to signs: Be careful with your signs, especially when dealing with negative numbers. A small sign error can lead to a completely wrong answer.
• Use a calculator to check your work: A calculator with complex number capabilities can be a valuable tool for verifying your answers. While it's essential to understand the steps involved in solving the problem, a calculator can help you catch any errors.
• Visualize complex numbers: Complex numbers can be represented graphically on a complex plane. Visualizing them can sometimes make it easier to understand their properties and operations.

Conclusion

Working with complex numbers might seem a bit challenging at first, but with a solid understanding of the basic concepts and a little practice, you'll be solving these problems like a pro in no time. Remember to break down the problem into smaller, manageable steps, and don't be afraid to ask for help if you get stuck. Keep practicing, and you'll unlock the power of complex numbers!

So, guys, hopefully, this step-by-step guide has helped you understand how to perform the indicated operations and express the answer in the form a + bi. Keep exploring the fascinating world of mathematics, and you'll be amazed at what you can discover!