Simplify -5-√-44: A Step-by-Step Guide

Hey guys! Today, we're going to tackle a fun little math problem: simplifying the expression $-5-\sqrt{-44}$. 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. So, grab your pencils and let's dive in!

Understanding the Problem

Before we jump into solving, let's make sure we understand what the problem is asking. We have the expression $-5-\sqrt{-44}$. The key here is the square root of a negative number, which introduces us to the world of imaginary numbers. Remember, the square root of a negative number involves the imaginary unit 'i', where $i = \sqrt{-1}$. This is super important, so keep it in mind as we move forward.

Now, let's identify the main parts of our expression. We have a real number, -5, and an imaginary part, $\sqrt{-44}$. Our goal is to simplify the imaginary part and combine it with the real part. Think of it like mixing ingredients in a recipe – we need to prepare each ingredient before we can combine them to get the final dish. In this case, we're preparing the $\sqrt{-44}$ part so we can combine it with -5 in the simplest way possible. We will simplify the square root of -44 and express it in terms of 'i'. This involves factoring out the perfect square from 44 and dealing with the negative sign inside the square root. Once we've simplified $\sqrt{-44}$, we'll subtract it from -5. This will give us our final answer in the form of a complex number, which is a number that has both a real part and an imaginary part.

So, to recap, we're dealing with complex numbers, which have both real and imaginary components. The imaginary unit 'i' is crucial here because it allows us to handle the square root of negative numbers. We'll simplify the expression by first simplifying the square root of -44 and then combining it with the real number -5. This will give us a complex number in its simplest form. Remember, math is like a puzzle, and each step is a piece that fits together to reveal the solution. Let's put those pieces together now!

Step-by-Step Solution

Okay, let's get into the nitty-gritty of solving this problem. Here's how we'll simplify $-5-\sqrt{-44}$ step by step:

Step 1: Factor out the negative sign

The first thing we need to do is deal with that negative sign inside the square root. Remember, $\sqrt{-1}$ is equal to 'i'. So, we can rewrite $\sqrt{-44}$ as $\sqrt{-1 * 44}$. This is a crucial step because it allows us to separate the imaginary part from the real number inside the square root. By factoring out the -1, we can use the definition of 'i' to simplify the expression further. This is like breaking down a complex problem into smaller, more manageable parts. Once we've separated the imaginary unit, we can focus on simplifying the remaining square root. This will make the overall process much easier.

Step 2: Factor the number inside the square root

Now, let's focus on the number 44. We want to find the largest perfect square that divides 44. This will help us simplify the square root. Think of perfect squares like 4, 9, 16, 25, and so on – numbers that are the result of squaring an integer. In this case, 44 can be factored as 4 * 11. And guess what? 4 is a perfect square! So, we can rewrite $\sqrt{44}$ as $\sqrt{4 * 11}$. This step is all about finding the right factors to make our simplification easier. By identifying the perfect square factor, we can take its square root and move it outside the radical. This leaves us with a smaller number inside the square root, which is much simpler to deal with.

Step 3: Simplify the square root

Using the property $\sqrt{a * b} = \sqrt{a} * \sqrt{b}$, we can rewrite $\sqrt{4 * 11}$ as $\sqrt{4} * \sqrt{11}$. We know that $\sqrt{4}$ is 2, so we now have 2 * $\sqrt{11}$. This is a big step forward! We've taken a potentially messy square root and simplified it into something much cleaner. This property of square roots is super handy for breaking down complex radicals into simpler parts. By applying this rule, we can often extract perfect square factors and simplify the expression significantly. In our case, it allowed us to turn $\sqrt{44}$ into 2$\sqrt{11}$, which is much easier to work with.

Step 4: Combine the imaginary unit

Remember we had $\sqrt{-1 * 44}$? We can now rewrite this as $\sqrt{-1} * \sqrt{44}$. We know $\sqrt{-1}$ is 'i', and we just simplified $\sqrt{44}$ to 2$\sqrt{11}$. So, $\sqrt{-44}$ becomes i * 2$\sqrt{11}$, which is usually written as 2i$\sqrt{11}$. This is where the imaginary unit 'i' comes back into play. We've successfully isolated the imaginary part of the expression and simplified it. By combining 'i' with the simplified square root, we've expressed the imaginary component in its standard form. This step is crucial for getting our final answer in the correct format for a complex number.

Step 5: Substitute back into the original expression

Now we substitute 2i$\sqrt{11}$ back into our original expression: $-5 - \sqrt{-44}$. This gives us $-5 - 2i\sqrt{11}$. And guess what? We're done! We've simplified the expression as much as possible. This final step is all about putting the pieces together. We've simplified the square root of -44, and now we're combining it with the real part of the expression. This gives us a complex number in its simplest form, with a real part and an imaginary part. It's like the grand finale of our math problem, where everything comes together to give us the final answer.

Final Answer

So, the simplified form of $-5-\sqrt{-44}$ is $-5 - 2i\sqrt{11}$. Looking at our options, the correct answer is (C). Great job, guys! You've successfully navigated the world of imaginary numbers and simplified a complex expression. Remember, the key is to break down the problem into smaller, manageable steps and take it one step at a time. You got this!

Why Other Options are Incorrect

Let's quickly look at why the other options are incorrect. This can help solidify our understanding and prevent future mistakes. It's always a good idea to understand not just why the correct answer is right, but also why the incorrect answers are wrong. This can give us a deeper understanding of the concepts involved and help us avoid common pitfalls.

Option A: $-5-4 \sqrt{11 i}$

This option is incorrect because it incorrectly places the 'i' inside the square root. Remember, the 'i' comes from the square root of -1, which we factored out. It should be outside the square root. This is a common mistake, so it's important to keep track of where the 'i' comes from and how it should be placed in the final expression. The 'i' represents the imaginary unit and should be treated separately from the real numbers inside the square root.

Option B: $-5-4 i \sqrt{11}$

This option has an incorrect coefficient for the imaginary term. We correctly factored out the perfect square 4 from 44, but the square root of 4 is 2, not 4. So, the coefficient should be 2, not 4. This highlights the importance of carefully simplifying the square root and paying attention to the coefficients. It's easy to make a mistake in the arithmetic, so it's always a good idea to double-check your work.

Option D: $-5-2 \sqrt{11 i}$

Similar to option A, this option incorrectly places the 'i' inside the square root. The 'i' should be outside the square root, as it represents the imaginary unit that we factored out. This mistake shows a misunderstanding of how the imaginary unit interacts with the square root. Remember, the 'i' is a separate entity that represents the square root of -1, and it should be treated accordingly.

By understanding why these options are incorrect, we reinforce our understanding of the correct method and the underlying concepts. It's like learning from our mistakes and building a stronger foundation for future problem-solving.

Practice Problems

Want to become a pro at simplifying complex numbers? Here are a few practice problems you can try:

1. Simplify $-3 + \sqrt{-25}$
2. Simplify $2 - \sqrt{-72}$
3. Simplify $-1 + \sqrt{-98}$

Remember to follow the same steps we used in the example problem: factor out the negative sign, find the largest perfect square factor, simplify the square root, and combine the imaginary unit. The more you practice, the more comfortable you'll become with these types of problems. Math is like a muscle – the more you exercise it, the stronger it gets!

Conclusion

Great job, guys! You've learned how to simplify expressions with imaginary numbers. Remember the key steps: factoring out the negative sign, simplifying the square root, and combining the real and imaginary parts. Keep practicing, and you'll be a complex number whiz in no time! Math can be challenging, but it's also incredibly rewarding when you finally solve a problem. So, keep up the great work and never stop learning!