Simplifying Complex Numbers: $\sqrt{-108}-\sqrt{-3}$ Explained

Hey math enthusiasts! Let's dive into a cool problem involving complex numbers. We're going to break down the expression $\sqrt{-108} - \sqrt{-3}$ and figure out which one is equivalent. Don't worry, it's not as scary as it looks. We'll go through the steps together, making sure everything is super clear and easy to follow. Complex numbers are numbers that can be expressed in the form of a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, defined as the square root of -1. This means i = √-1. Understanding this is key to solving our problem.

Understanding Complex Numbers and the Imaginary Unit

Complex numbers are the kind of numbers that extend the real number system by including the imaginary unit, often denoted as 'i'. This 'i' is super important because it's defined as the square root of -1 (√-1). So, whenever we see a negative number under a square root, we know we're dealing with complex numbers. For example, expressions like √-4, √-9, or in our case, √-108 and √-3, all involve complex numbers. This is because they can be rewritten using 'i'.

To really get this, imagine you're trying to find a number that, when multiplied by itself, gives you -1. This is where the magic of the imaginary unit comes in. There's no real number that can do this, but 'i' can. This 'i' opens up a whole new world of numbers, allowing us to deal with square roots of negative numbers without running into a dead end. Therefore, when we simplify expressions with negative square roots, we pull out the 'i' and then work with the remaining real numbers. For instance, √-4 simplifies to 2i (because √-4 = √(4 * -1) = √4 * √-1 = 2i), and √-9 simplifies to 3i (because √-9 = √(9 * -1) = √9 * √-1 = 3i). In our main problem, we will use this concept to simplify √-108 and √-3. This initial move sets the stage for a straightforward solution. So, let’s get started with simplifying √-108 and √-3, so we can finally subtract them.

Now, let's look at how we can rewrite the terms in our original expression. This involves breaking down the negative numbers under the square root and using the property of 'i' that we just discussed. This is how we can simplify √-108 and √-3 and proceed with the expression: $\sqrt{-108} - \sqrt{-3}$.

Breaking Down the Expression: $\sqrt{-108}$

Alright, let's start with $\sqrt{-108}$. The goal here is to rewrite this in a way that separates the negative part from the positive part, allowing us to use the imaginary unit 'i'. Here's how we do it: First, we can rewrite $\sqrt{-108}$ as $\sqrt{108 * -1}$. This doesn't change the value, but it does help us see the components we need. Next, we can separate the square root into two parts: $\sqrt{108} * \sqrt{-1}$. We know that $\sqrt{-1}$ is equal to 'i', so we're making progress. Now, we need to simplify $\sqrt{108}$.

To simplify $\sqrt{108}$, we look for perfect square factors of 108. The largest perfect square factor of 108 is 36 (because 36 * 3 = 108). So, we can rewrite $\sqrt{108}$ as $\sqrt{36 * 3}$. Now, using the property of square roots, we can separate this into $\sqrt{36} * \sqrt{3}$. We know that $\sqrt{36}$ is 6, so we simplify this part to 6$\sqrt{3}$.

Putting it all together, $\sqrt{-108}$ becomes 6$\sqrt{3}$ * i, or simply 6i$\sqrt{3}$. So, we have successfully simplified the first term. This might sound a bit complex at first, but with practice, you will find it becomes much easier. It's like finding a secret code: once you know how to crack it, you can solve similar problems quickly. Always remember to look for the largest perfect square factor to simplify the square root effectively. This approach makes sure our work stays neat and easy to understand. Now that we have handled $\sqrt{-108}$, we move on to the second part of our expression, $\sqrt{-3}$.

Simplifying $\sqrt{-3}$

Now, let's take a look at the second part of our expression, which is $\sqrt{-3}$. Similar to what we did with $\sqrt{-108}$, we want to rewrite this term using the imaginary unit 'i'. The process is pretty similar to the first part, but with slightly different numbers. Here's what we do. We begin by rewriting $\sqrt{-3}$ as $\sqrt{3 * -1}$. This separates the positive part (3) from the negative part (-1). Then, we can separate the square root into two parts: $\sqrt{3} * \sqrt{-1}$. And once again, $\sqrt{-1}$ is 'i'.

Therefore, $\sqrt{-3}$ simplifies to i$\sqrt{3}$. Here, the $\sqrt{3}$ stays as is because 3 doesn't have any perfect square factors other than 1. This means we can't simplify it further. It's crucial to notice that sometimes the square root remains in the answer, as is the case here. Understanding how to handle these situations is an important part of solving complex number problems. This step is much simpler than the first one. By breaking it down step by step, it makes the math easier to handle. Now, we have both $\sqrt{-108}$ and $\sqrt{-3}$ in their simplest forms, which allows us to return to the original problem. Let’s finish this! We are almost there.

Putting It All Together: Solving $\sqrt{-108} - \sqrt{-3}$

Okay, guys, we're at the finish line! Now that we've simplified both $\sqrt{-108}$ to 6i$\sqrt{3}$ and $\sqrt{-3}$ to i$\sqrt{3}$, we can go back to the original problem: $\sqrt{-108} - \sqrt{-3}$. We already know that $\sqrt{-108}$ is equal to 6i$\sqrt{3}$ and $\sqrt{-3}$ is equal to i$\sqrt{3}$. So, we can rewrite the expression as 6i$\sqrt{3}$ - i$\sqrt{3}$.

Now, we just need to subtract the two terms. Since both terms have i$\sqrt{3}$ in them, we can treat it like a common factor. Imagine it like this: 6 apples - 1 apple = 5 apples. The same logic applies here. So, 6i$\sqrt{3}$ - i$\sqrt{3}$ equals 5i$\sqrt{3}$. That’s it! That’s our answer.

Therefore, the expression $\sqrt{-108} - \sqrt{-3}$ simplifies to 5i$\sqrt{3}$.

This entire process may seem long, but each step is necessary to accurately solve the problem. The most important thing to remember is the definition of 'i' and how it allows us to work with negative square roots. Also, remember to look for the largest perfect square factor to simplify the square roots efficiently. If you get stuck, always break the problem down into smaller parts. With practice, you’ll become more comfortable with these types of problems. Remember, math is about learning, so don’t hesitate to ask for help or to practice more problems! Great job! You successfully found the solution.

Final Answer

The equivalent expression is 5i$\sqrt{3}$.