Simplifying $(\sqrt{3}+i)^3$ A Step-by-Step Guide

Hey guys! Today, we're diving deep into the fascinating world of complex numbers and tackling a problem that might seem a bit intimidating at first glance: simplifying the expression $(\sqrt{3}+i)^3$. But don't worry, we'll break it down step-by-step, making sure everyone understands the process. So, grab your thinking caps, and let's get started!

Understanding Complex Numbers and Their Representations

Before we jump into the calculation, let's refresh our understanding of complex numbers. A complex number is essentially a number that can be expressed in the form $a + bi$, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, defined as $i = \sqrt{-1}$. The 'a' part is called the real part, and the 'b' part is called the imaginary part. Now, complex numbers aren't just abstract concepts; they have a beautiful geometric interpretation on the complex plane. Think of it as a regular Cartesian plane, but instead of the x-axis representing real numbers and the y-axis representing imaginary numbers. A complex number $a + bi$ can be plotted as a point (a, b) on this plane.

But that's not all! Complex numbers can also be represented in polar form. This is where things get interesting for our problem. Polar form expresses a complex number in terms of its magnitude (or modulus) 'r' and its angle (or argument) 'θ'. The magnitude 'r' is the distance from the origin (0, 0) to the point representing the complex number on the complex plane, and it can be calculated using the Pythagorean theorem: $r = \sqrt{a^2 + b^2}$. The angle 'θ' is the angle formed between the positive real axis and the line connecting the origin to the point, and it can be found using trigonometric functions like $\tan θ = \frac{b}{a}$. So, a complex number $a + bi$ can be written in polar form as $r(\cos θ + i \sin θ)$. This representation is super handy when we need to multiply or raise complex numbers to a power, as we'll see shortly.

In our specific problem, we have the complex number $\sqrt{3} + i$. Here, $a = \sqrt{3}$ and $b = 1$. Let's find its polar form. The magnitude 'r' is $\sqrt{(\sqrt{3})^2 + 1^2} = \sqrt{3 + 1} = \sqrt{4} = 2$. The angle 'θ' can be found using $\tan θ = \frac{1}{\sqrt{3}}$. We know that $\tan(\frac{π}{6}) = \frac{1}{\sqrt{3}}$, so $θ = \frac{π}{6}$. Therefore, the polar form of $\sqrt{3} + i$ is $2(\cos(\frac{π}{6}) + i \sin(\frac{π}{6}))$. This conversion is crucial because it sets the stage for using De Moivre's Theorem, a powerful tool for handling powers of complex numbers.

De Moivre's Theorem: The Key to Unlocking the Power

Now, let's talk about De Moivre's Theorem. This theorem is a cornerstone when dealing with powers of complex numbers in polar form. It states that for any complex number in polar form $r(\cos θ + i \sin θ)$ and any integer n, the following holds true: $[r(\cos θ + i \sin θ)]^n = r^n(\cos(nθ) + i \sin(nθ))$. In simpler terms, to raise a complex number in polar form to a power, you raise the magnitude to that power and multiply the angle by that power. It's like a shortcut that saves us from tedious multiplications.

Why is this theorem so powerful? Well, imagine trying to expand $(\sqrt{3} + i)^3$ directly using the binomial theorem. It would involve a lot of calculations and potential for errors. But with De Moivre's Theorem, we can bypass all that. We've already converted $\sqrt{3} + i$ to its polar form: $2(\cos(\frac{π}{6}) + i \sin(\frac{π}{6}))$. Now, to find $(\sqrt{3} + i)^3$, we simply apply De Moivre's Theorem with n = 3. This means we raise the magnitude (2) to the power of 3 and multiply the angle $(\frac{π}{6})$ by 3. So, we get:

$[2(\cos(\frac{π}{6}) + i \sin(\frac{π}{6}))]^3 = 2^3(\cos(3 * \frac{π}{6}) + i \sin(3 * \frac{π}{6})) = 8(\cos(\frac{π}{2}) + i \sin(\frac{π}{2}))$. See how much simpler that was? De Moivre's Theorem transformed a complex calculation into a straightforward application of a formula. This is why understanding and utilizing this theorem is so vital when dealing with powers of complex numbers. It's not just about getting the right answer; it's about understanding the elegance and efficiency of mathematical tools.

Applying De Moivre's Theorem to Solve the Problem

Okay, let's get back to our original problem: simplifying $(\sqrt{3}+i)^3$. We've already laid the groundwork by converting $\sqrt{3} + i$ to its polar form and understanding De Moivre's Theorem. Now, it's time to put everything together and find the solution. As we established earlier, the polar form of $\sqrt{3} + i$ is $2(\cos(\frac{π}{6}) + i \sin(\frac{π}{6}))$. Applying De Moivre's Theorem with n = 3, we get:

$(\sqrt{3} + i)^3 = [2(\cos(\frac{π}{6}) + i \sin(\frac{π}{6}))]^3 = 2^3(\cos(3 * \frac{π}{6}) + i \sin(3 * \frac{π}{6})) = 8(\cos(\frac{π}{2}) + i \sin(\frac{π}{2}))$. So, we've simplified the expression to $8(\cos(\frac{π}{2}) + i \sin(\frac{π}{2}))$. But we're not quite done yet. To fully understand the result and match it with the answer choices, we need to evaluate the cosine and sine functions.

We know that $\cos(\frac{π}{2}) = 0$ and $\sin(\frac{π}{2}) = 1$. Substituting these values into our expression, we get:

$8(\cos(\frac{π}{2}) + i \sin(\frac{π}{2})) = 8(0 + i * 1) = 8i$. Aha! So, $(\sqrt{3} + i)^3$ simplifies to $8i$. This is a purely imaginary number, which makes sense given the angle we ended up with. Now, let's revisit the original options and see which one matches our simplified form.

Looking at the options, we can see that option A, $6[\cos(\frac{π}{2}) + i \sin(\frac{π}{2})]$, B, $6[\cos(\frac{π}{6}) + i \sin(\frac{π}{6})]$, and C, $8[\cos(\frac{π}{2}) + i \sin(\frac{π}{2})]$ is the correct answer. Option C perfectly matches our result, $8(\cos(\frac{π}{2}) + i \sin(\frac{π}{2}))$, which we know is equivalent to 8i. The other options have different magnitudes or angles, so they don't represent the same complex number. This methodical approach, using polar form and De Moivre's Theorem, allows us to confidently arrive at the correct answer. It's a powerful technique for simplifying complex expressions and a testament to the beauty and efficiency of mathematical tools.

Converting Back to Rectangular Form (a + bi)

While we've successfully simplified the expression and identified the correct answer, let's take a moment to solidify our understanding by converting our polar form result back to rectangular form (a + bi). This is a great way to double-check our work and ensure we truly grasp the concepts involved. We arrived at $8(\cos(\frac{π}{2}) + i \sin(\frac{π}{2}))$ as the simplified polar form of $(\sqrt{3} + i)^3$. To convert this back to rectangular form, we simply evaluate the cosine and sine and distribute the magnitude.

As we mentioned earlier, $\cos(\frac{π}{2}) = 0$ and $\sin(\frac{π}{2}) = 1$. So, our expression becomes:

$8(\cos(\frac{π}{2}) + i \sin(\frac{π}{2})) = 8(0 + i * 1) = 8(0 + i) = 8i$. This is the rectangular form of the complex number. We can write it as 0 + 8i, where the real part is 0 and the imaginary part is 8. This confirms our earlier finding that $(\sqrt{3} + i)^3$ is a purely imaginary number. The rectangular form provides a different perspective on the complex number, allowing us to visualize it as a point (0, 8) on the complex plane. It lies on the positive imaginary axis, 8 units away from the origin. This conversion process highlights the connection between polar and rectangular forms and demonstrates how we can seamlessly move between these representations to solve problems.

By converting back to rectangular form, we've not only verified our solution but also gained a deeper appreciation for the interplay between different representations of complex numbers. This flexibility in representation is what makes complex numbers so powerful and versatile in various applications, from electrical engineering to quantum mechanics. So, the next time you encounter a complex number problem, remember the tools we've discussed – polar form, De Moivre's Theorem, and the ability to convert between forms – and you'll be well-equipped to tackle it!

Conclusion: Mastering Complex Numbers

So, guys, we've successfully navigated the world of complex numbers and simplified the expression $(\sqrt{3}+i)^3$. We started by understanding the basics of complex numbers and their polar representation. Then, we harnessed the power of De Moivre's Theorem to efficiently raise the complex number to a power. Finally, we converted our result back to rectangular form to solidify our understanding. This journey highlights the importance of having a solid grasp of fundamental concepts and the ability to apply them strategically. Complex numbers might seem intimidating at first, but with the right tools and a step-by-step approach, they become much more manageable and even fascinating.

Remember, the key to mastering complex numbers lies in practice and understanding the underlying principles. Don't be afraid to tackle challenging problems and explore different approaches. The more you work with these concepts, the more comfortable and confident you'll become. And who knows, you might even start to see the beauty and elegance that complex numbers have to offer. Keep exploring, keep learning, and keep having fun with math!