Simplifying Complex Numbers Finding Equivalent Expression For $\sqrt{-108}-\sqrt{-3}$

When delving into the world of complex numbers, simplifying expressions involving square roots of negative numbers is a fundamental skill. In this article, we will dissect the expression $\sqrt{-108}-\sqrt{-3}$, providing a comprehensive, step-by-step guide to arrive at the correct equivalent expression. Our journey will not only reveal the solution but also enhance your understanding of complex number arithmetic. Let's embark on this mathematical adventure!

At the heart of complex numbers lies the imaginary unit, i, defined as the square root of -1. This seemingly simple concept unlocks a new dimension in mathematics, allowing us to work with the square roots of negative numbers. To simplify expressions like $\sqrt{-108}-\sqrt{-3}$, we must first recognize that the square root of a negative number can be expressed as a multiple of i. The imaginary unit i is the cornerstone of complex numbers. Understanding its properties is crucial for navigating the complex plane and performing various mathematical operations. We will first express the square roots of negative numbers in terms of i, and then reduce the radical, and only then will we be ready to solve the problem. This step allows us to rewrite $\sqrt{-108}$ and $\sqrt{-3}$ in a more manageable form, setting the stage for simplification.

$\sqrt{-108}$ can be rewritten as $\sqrt{108 \cdot -1}$. Now we use the property of radicals that says $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$. Therefore, we can rewrite this as $\sqrt{108} \cdot \sqrt{-1}$. And we know that $\sqrt{-1}$ is i. We can rewrite this as $i\sqrt{108}$. Let's work on simplifying the $\sqrt{108}$. We want to find the largest perfect square that divides 108. We can make a prime factorization tree to help us. $108 = 2 \cdot 54$, $54 = 2 \cdot 27$, $27 = 3 \cdot 9$, and $9 = 3 \cdot 3$. So we have that $108 = 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 = 2^2 \cdot 3^2 \cdot 3$. Now we can rewrite our expression with these factors, $i\sqrt{2^2 \cdot 3^2 \cdot 3}$. Again, using the property of radicals, we can write this as $i \cdot \sqrt{2^2} \cdot \sqrt{3^2} \cdot \sqrt{3}$. Now we can simplify the square roots. $\sqrt{2^2} = 2$ and $\sqrt{3^2} = 3$. So our expression becomes $i \cdot 2 \cdot 3 \cdot \sqrt{3}$ which equals $6i\sqrt{3}$.

$\sqrt{-3}$ can be rewritten as $\sqrt{3 \cdot -1}$. Again, we use the property of radicals to rewrite this as $\sqrt{3} \cdot \sqrt{-1}$. Since $\sqrt{-1} = i$, we can rewrite our expression as $i\sqrt{3}$. Notice in this case that 3 is a prime number, so we have simplified the radical as much as we can.

With the individual terms simplified, the next step is to perform the subtraction. We are tasked with finding the equivalent expression for $\sqrt{-108}-\sqrt{-3}$. Having simplified each term, we now have $6i\sqrt{3} - i\sqrt{3}$. Now we need to combine like terms. In this case, we can factor out the common term of $i\sqrt{3}$. This gives us $(6-1)i\sqrt{3}$. We can simplify this to $5i\sqrt{3}$. This is our final answer.

Through careful simplification and combination of like terms, we have determined that the expression $\sqrt{-108}-\sqrt{-3}$ is equivalent to $5i\sqrt{3}$. Therefore, the correct answer is:

A. $5i\sqrt{3}$

Simplifying complex number expressions can seem daunting at first, but by following a systematic approach, it becomes a manageable task. Let's solidify your understanding with a recap of the essential steps and concepts:

1. Expressing Square Roots of Negatives in Terms of i

The imaginary unit i is the key to unlocking the world of complex numbers. Remember that $\sqrt{-1} = i$, and this allows you to rewrite the square root of any negative number in terms of i. For example, $\sqrt{-a} = i\sqrt{a}$, where a is a positive real number. This first step is crucial for making the expression workable.

2. Simplifying Radicals: Finding Perfect Square Factors

Once you've expressed the square root in terms of i, the next step is to simplify the radical. This involves identifying the largest perfect square that divides the number under the radical. Factoring out perfect squares allows you to reduce the radical to its simplest form. In our example, we broke down $\sqrt{108}$ by finding its prime factors and identifying pairs, which correspond to perfect squares.

3. Combining Like Terms: Treating i as a Variable

When performing addition or subtraction with complex numbers, you treat i like a variable. Combine the terms that have the same radical and the imaginary unit. This step ensures that you are only combining terms that are mathematically compatible.

4. Attention to Detail: Avoiding Common Pitfalls

Accuracy is paramount in mathematics. Double-check your work, especially when simplifying radicals and combining terms. A small error in one step can lead to an incorrect final answer. Be methodical in your approach, and don't rush through the process.

Like any mathematical skill, mastering complex number simplification requires practice. Challenge yourself with a variety of problems, and don't be afraid to seek help when needed. The more you practice, the more confident and proficient you will become. This article has equipped you with the tools and knowledge to tackle similar problems. Remember to break down complex expressions into manageable steps, and you'll be well on your way to success.

Simplifying complex number expressions is not just about finding the right answer; it's about developing a deeper understanding of mathematical principles. By following a systematic approach, you can transform seemingly complicated problems into manageable tasks. The beauty of mathematics lies in its ability to reveal order and simplicity in the face of complexity. Keep exploring, keep practicing, and keep unlocking the wonders of mathematics.