Let's break down the expression and rewrite it in its rectangular form, which is in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit ($i = \sqrt{-1}$).

The expression is:

$-5^{7} + 4i^{12} + (\sqrt{-9})^{3}$

Step-by-Step Breakdown:

1. Term 1: $-5^7$
   This term involves a real number raised to a power, so it is straightforward. $-5^7 = -78125$

2. Term 2: $4i^{12}$
   We know that powers of $i$ follow a cyclical pattern: $i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1, \quad \text{and then it repeats}.$ Since $12 \div 4 = 3$ (i.e., $i^{12} = (i^4)^3 = 1^3 = 1$), we have: $4i^{12} = 4 \times 1 = 4$

3. Term 3: $(\sqrt{-9})^3$
   The square root of $-9$ is: $\sqrt{-9} = 3i$ Now, we cube it: $(3i)^3 = 3^3 \times i^3 = 27 \times (-i) = -27i$

Putting it all together:

Now, we can combine the real and imaginary parts: $-78125 + 4 + (-27i) = -78121 - 27i$

So, the rectangular form of the expression is: $\boxed{-78121 - 27i}$

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Would you like further clarification on any of the steps? Or do you have any other math questions?

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Related Questions:

1. How do powers of $i$ work? Can you show how to simplify higher powers of $i$?
2. How can we express complex numbers in polar or exponential form?
3. What is the significance of complex conjugates, and how do they relate to rectangular form?
4. How would you add or subtract complex numbers in rectangular form?
5. Can you explain how to multiply and divide complex numbers?

Tip: Always reduce powers of $i$ by using the cyclical pattern $i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1$, and repeat from there!