Simplify $-6 oldsymbol{\sqrt{-20}}$ To $a+bi$

Hey guys, ever stared at a math problem and thought, "What in the world is this asking?" Well, you're not alone! Today, we're diving deep into the funky world of complex numbers to tackle a question that might seem a bit intimidating at first glance. We're going to figure out how to re-express the negative square root $-6 \sqrt{-20}$ as a complex number in the standard form $a+bi$. So grab your calculators, maybe a comfy seat, and let's break this down.

Understanding the Negative Square Root and Complex Numbers

Alright, first things first, let's talk about what we're dealing with. We've got $-6 \sqrt{-20}$. The negative sign outside the square root tells us we're looking for the negative square root of whatever is inside. The real kicker here is the $-20$ under the square root. You know how in the real number system, you can't take the square root of a negative number? That's where our buddies, the complex numbers, come to the rescue!

Complex numbers are built upon the idea that there's a special number, denoted by '$i$', which is defined as the square root of $-1$. So, $i = \sqrt{-1}$, and importantly, $i^2 = -1$. This little '$i$' unlocks a whole new realm of mathematics, allowing us to work with the square roots of negative numbers. The standard form of a complex number is $a+bi$, where '$a$' is the real part and '$b$' is the imaginary part. Our mission, should we choose to accept it, is to manipulate $-6 \sqrt{-20}$ until it fits this $a+bi$ format. It's like giving a number a makeover so it looks nice and tidy in its standard outfit.

Breaking Down the Square Root of -20

Now, let's focus on that tricky part: $\sqrt{-20}$. To deal with the negative inside the square root, we can use our definition of '$i$'. We can rewrite $\sqrt{-20}$ as $\sqrt{20 \times -1}$. Using the property of square roots that $\sqrt{xy} = \sqrt{x} \times \sqrt{y}$, we can split this up: $\sqrt{20} \times \sqrt{-1}$. We know that $\sqrt{-1}$ is just '$i$', so we have $\sqrt{20} \times i$.

But wait, there's more! We can simplify $\sqrt{20}$ further. Think about the factors of 20. We're looking for a perfect square factor. Aha! $20 = 4 \times 5$, and 4 is a perfect square ($2^2$). So, $\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2 \sqrt{5}$.

Putting it all together, $\sqrt{-20}$ simplifies to $2 \sqrt{5} \times i$, or more commonly written as $2i\sqrt{5}$. This is the simplified form of the square root of $-20$, incorporating the imaginary unit '$i$' and simplifying the radical part. So, whenever you see a negative under a square root, just pull out the '$i$' and deal with the positive part separately. It's like extracting the imaginary essence and leaving the real radical behind!

Incorporating the -6 Multiplier

Okay, we've wrestled $\sqrt{-20}$ into submission, getting $2i\sqrt{5}$. Now, let's bring back that $-6$ that was chilling outside the original square root. Our original expression was $-6 \sqrt{-20}$. We're going to substitute our simplified version of $\sqrt{-20}$ back into the equation. So, we have $-6 \times (2i\sqrt{5})$.

Multiplying these together is pretty straightforward. We multiply the coefficients (the numbers in front). We have $-6 \times 2$, which gives us $-12$. The '$i$' and the '$\sqrt{5}$' just tag along for the ride. So, the result of the multiplication is $-12i\sqrt{5}$.

At this point, we should pause and check if this result fits the standard form of a complex number, which is $a+bi$. Our current result, $-12i\sqrt{5}$, can be thought of as $0 + (-12\sqrt{5})i$. Here, '$a$' (the real part) is 0, and '$b$' (the coefficient of '$i$') is $-12\sqrt{5}$. So, yes, it is a complex number, and it's expressed in a way that directly relates to the $a+bi$ format. This is the final, simplified form of the original expression.

Matching with the Options

Now for the final step, which is always satisfying: comparing our hard-earned answer with the given options. We simplified $-6 \sqrt{-20}$ and arrived at $-12i\sqrt{5}$. Let's look at the choices provided:

A. $-12 \sqrt{5}$ B. $2 i \sqrt{5}$ C. $-12 i \sqrt{5}$ D. $-12 \sqrt{5 i}$

Looking at our result, $-12i\sqrt{5}$, we can see it directly matches option C. Option A, $-12\sqrt{5}$, is incorrect because it's missing the imaginary unit '$i$', meaning it's a purely real number. Option B, $2i\sqrt{5}$, is the simplified version of $\sqrt{-20}$ itself, but it doesn't include the multiplier of $-6$. Option D, $-12\sqrt{5i}$, is structured differently and implies the imaginary unit is inside the square root in a way that doesn't match our calculation. It's crucial to keep track of every part of the original expression throughout the simplification process.

Therefore, the correct re-expression of $-6 \sqrt{-20}$ as a complex number in the standard form $a+bi$ is $-12i\sqrt{5}$. It's awesome how we can transform seemingly complicated expressions into a clear, standard format using the rules of algebra and the magic of imaginary numbers. Keep practicing, guys, and these kinds of problems will become second nature!