Finding Square Roots Of Complex Numbers $-8+8 I \sqrt{3}$

Hey guys! Today, we're diving into a fascinating mathematical puzzle: finding the square roots of the complex number $-8 + 8i\[sqrt{3}]$. This might sound intimidating, but don't worry, we'll break it down step by step. Complex numbers, with their real and imaginary parts, can be a bit tricky, but once you grasp the underlying principles, you'll find it's actually quite fun! We're given a list of potential square roots, and our mission is to verify which ones actually fit the bill. So, let's put on our thinking caps and get started!

Understanding Complex Numbers and Square Roots

Before we jump into the calculations, let's quickly recap what complex numbers are and what it means to find their square roots. Complex numbers, at their core, are numbers that can be expressed in the form $a + bi$, where 'a' is the real part, 'b' is the imaginary part, and 'i' is the imaginary unit, defined as the square root of -1 (i.e., $i = \sqrt{-1}$). Think of them as extending the regular number line into a two-dimensional plane, where the x-axis represents the real part and the y-axis represents the imaginary part. This plane is often referred to as the complex plane or Argand diagram. Visualizing complex numbers in this way can make operations like addition, subtraction, multiplication, and division more intuitive. When we talk about finding the square root of a complex number, we're essentially asking: What complex number, when multiplied by itself, gives us the original complex number? In other words, if we have a complex number $z$, we're looking for another complex number $w$ such that $w^2 = z$. This is similar to finding the square root of a real number, but with the added complexity of dealing with imaginary components. The beauty of complex numbers lies in their ability to elegantly solve problems that are intractable within the realm of real numbers alone. For example, equations like $x^2 + 1 = 0$ have no real solutions, but they have two complex solutions: $x = i$ and $x = -i$. This opens up a whole new world of mathematical possibilities and applications, from electrical engineering to quantum mechanics.

The Challenge: Square Roots of -8 + 8i√3

Now, let's get back to our specific problem. We're tasked with identifying the square roots of the complex number $-8 + 8i\[sqrt{3}]$. This means we need to find complex numbers that, when squared, result in $-8 + 8i\[sqrt{3}]$. The list of potential square roots we have includes:

• $-2 + 2i\[sqrt{3}]$
• $-2 - 2i\[sqrt{3}]$
• $2 + 2i\[sqrt{3}]$
• $-4 + 4i\[sqrt{3}]$
• $4 - 4i\[sqrt{3}]$
• $4 + 4i\[sqrt{3}]$

Our strategy will be straightforward: we'll square each of these complex numbers and see if the result matches our target, $-8 + 8i\[sqrt{3}]$. This is a bit like checking if a key fits a lock – we try each key until we find the ones that work. Remember, squaring a complex number involves multiplying it by itself, which means we'll be using the distributive property (often remembered by the acronym FOIL – First, Outer, Inner, Last) and the fact that $i^2 = -1$. This process will not only help us identify the correct square roots but also deepen our understanding of how complex numbers interact with each other under multiplication. So, let's roll up our sleeves and start the verification process!

Verification Process: Squaring the Potential Roots

Let's take each potential root and square it to see if we get $-8 + 8i\[sqrt{3}]$. This is the heart of our problem-solving strategy, and it involves careful application of complex number multiplication rules. We'll go through each option methodically, showing the steps involved so you can follow along and understand the process. Remember, the key is to treat 'i' as a variable but also keep in mind that $i^2 = -1$. This seemingly simple rule is what allows us to combine real and imaginary terms after the multiplication is done.

1. Checking -2 + 2i√3

To check the first option, $-2 + 2i\[sqrt{3}]$, we need to square it:

$(-2 + 2i\[sqrt{3}])^2 = (-2 + 2i\[sqrt{3}])(-2 + 2i\[sqrt{3}])$

Now, we apply the distributive property (FOIL):

• First: $(-2) * (-2) = 4$
• Outer: $(-2) * (2i\[sqrt{3}]) = -4i\[sqrt{3}]$
• Inner: $(2i\[sqrt{3}]) * (-2) = -4i\[sqrt{3}]$
• Last: $(2i\[sqrt{3}]) * (2i\[sqrt{3}]) = 4i^2 * 3 = 12i^2$

Combining these, we get:

$4 - 4i\[sqrt{3}] - 4i\[sqrt{3}] + 12i^2$

Remember that $i^2 = -1$, so we can substitute that in:

$4 - 8i\[sqrt{3}] + 12(-1) = 4 - 8i\[sqrt{3}] - 12 = -8 - 8i\[sqrt{3}]$

This does not match $-8 + 8i\[sqrt{3}]$, so $-2 + 2i\[sqrt{3}]$ is not a square root.

2. Checking -2 - 2i√3

Next, let's check $-2 - 2i\[sqrt{3}]$

$(-2 - 2i\[sqrt{3}])^2 = (-2 - 2i\[sqrt{3}])(-2 - 2i\[sqrt{3}])$

Applying the distributive property:

• First: $(-2) * (-2) = 4$
• Outer: $(-2) * (-2i\[sqrt{3}]) = 4i\[sqrt{3}]$
• Inner: $(-2i\[sqrt{3}]) * (-2) = 4i\[sqrt{3}]$
• Last: $(-2i\[sqrt{3}]) * (-2i\[sqrt{3}]) = 4i^2 * 3 = 12i^2$

Combining these, we get:

$4 + 4i\[sqrt{3}] + 4i\[sqrt{3}] + 12i^2$

Substituting $i^2 = -1$:

$4 + 8i\[sqrt{3}] + 12(-1) = 4 + 8i\[sqrt{3}] - 12 = -8 + 8i\[sqrt{3}]$

This matches $-8 + 8i\[sqrt{3}]$, so $-2 - 2i\[sqrt{3}]$ is a square root!

3. Checking 2 + 2i√3

Now, let's check $2 + 2i\[sqrt{3}]$

$(2 + 2i\[sqrt{3}])^2 = (2 + 2i\[sqrt{3}])(2 + 2i\[sqrt{3}])$

Applying the distributive property:

• First: $(2) * (2) = 4$
• Outer: $(2) * (2i\[sqrt{3}]) = 4i\[sqrt{3}]$
• Inner: $(2i\[sqrt{3}]) * (2) = 4i\[sqrt{3}]$
• Last: $(2i\[sqrt{3}]) * (2i\[sqrt{3}]) = 4i^2 * 3 = 12i^2$

Combining these, we get:

$4 + 4i\[sqrt{3}] + 4i\[sqrt{3}] + 12i^2$

Substituting $i^2 = -1$:

$4 + 8i\[sqrt{3}] + 12(-1) = 4 + 8i\[sqrt{3}] - 12 = -8 + 8i\[sqrt{3}]$

This matches $-8 + 8i\[sqrt{3}]$, so $2 + 2i\[sqrt{3}]$ is a square root!

4. Checking -4 + 4i√3

Let's check $-4 + 4i\[sqrt{3}]$

$(-4 + 4i\[sqrt{3}])^2 = (-4 + 4i\[sqrt{3}])(-4 + 4i\[sqrt{3}])$

Applying the distributive property:

• First: $(-4) * (-4) = 16$
• Outer: $(-4) * (4i\[sqrt{3}]) = -16i\[sqrt{3}]$
• Inner: $(4i\[sqrt{3}]) * (-4) = -16i\[sqrt{3}]$
• Last: $(4i\[sqrt{3}]) * (4i\[sqrt{3}]) = 16i^2 * 3 = 48i^2$

Combining these, we get:

$16 - 16i\[sqrt{3}] - 16i\[sqrt{3}] + 48i^2$

Substituting $i^2 = -1$:

$16 - 32i\[sqrt{3}] + 48(-1) = 16 - 32i\[sqrt{3}] - 48 = -32 - 32i\[sqrt{3}]$

This does not match $-8 + 8i\[sqrt{3}]$, so $-4 + 4i\[sqrt{3}]$ is not a square root.

5. Checking 4 - 4i√3

Next, we'll check $4 - 4i\[sqrt{3}]$

$(4 - 4i\[sqrt{3}])^2 = (4 - 4i\[sqrt{3}])(4 - 4i\[sqrt{3}])$

Applying the distributive property:

• First: $(4) * (4) = 16$
• Outer: $(4) * (-4i\[sqrt{3}]) = -16i\[sqrt{3}]$
• Inner: $(-4i\[sqrt{3}]) * (4) = -16i\[sqrt{3}]$
• Last: $(-4i\[sqrt{3}]) * (-4i\[sqrt{3}]) = 16i^2 * 3 = 48i^2$

Combining these, we get:

$16 - 16i\[sqrt{3}] - 16i\[sqrt{3}] + 48i^2$

Substituting $i^2 = -1$:

$16 - 32i\[sqrt{3}] + 48(-1) = 16 - 32i\[sqrt{3}] - 48 = -32 - 32i\[sqrt{3}]$

This does not match $-8 + 8i\[sqrt{3}]$, so $4 - 4i\[sqrt{3}]$ is not a square root.

6. Checking 4 + 4i√3

Finally, let's check $4 + 4i\[sqrt{3}]$

$(4 + 4i\[sqrt{3}])^2 = (4 + 4i\[sqrt{3}])(4 + 4i\[sqrt{3}])$

Applying the distributive property:

• First: $(4) * (4) = 16$
• Outer: $(4) * (4i\[sqrt{3}]) = 16i\[sqrt{3}]$
• Inner: $(4i\[sqrt{3}]) * (4) = 16i\[sqrt{3}]$
• Last: $(4i\[sqrt{3}]) * (4i\[sqrt{3}]) = 16i^2 * 3 = 48i^2$

Combining these, we get:

$16 + 16i\[sqrt{3}] + 16i\[sqrt{3}] + 48i^2$

Substituting $i^2 = -1$:

$16 + 32i\[sqrt{3}] + 48(-1) = 16 + 32i\[sqrt{3}] - 48 = -32 + 32i\[sqrt{3}]$

This does not match $-8 + 8i\[sqrt{3}]$, so $4 + 4i\[sqrt{3}]$ is not a square root.

Conclusion: The True Square Roots Revealed

After meticulously squaring each potential root and comparing the results, we've successfully identified the square roots of $-8 + 8i\[sqrt{3}]$. It turns out that only two of the options given actually work:

• -2 - 2i√3
• 2 + 2i√3

It's interesting to note that these two square roots are negatives of each other, which is a common characteristic of square roots in both real and complex number systems. This exercise not only helped us find the correct answers but also reinforced our understanding of complex number arithmetic, particularly the multiplication of complex numbers. Remember, when dealing with complex numbers, the key is to be patient, methodical, and to keep in mind the fundamental property of the imaginary unit: $i^2 = -1$. With these tools in hand, you'll be well-equipped to tackle a wide range of complex number problems!

So there you have it, guys! We've successfully decoded the square roots of $-8 + 8i\[sqrt{3}]$. I hope this breakdown has been helpful and has shed some light on the fascinating world of complex numbers. Keep practicing, and you'll become a complex number whiz in no time!