$$ \sqrt{8\left(4092\right)-\left(36864\right)} $$

Let's solve the given expression:

\[
\sqrt{8(4092) - 36864}
\]

### Step 1: Multiply inside the square root

First, multiply $ 8 $ by $ 4092 $:

\[
8 \times 4092 = 32736
\]

### Step 2: Subtract

Now, subtract $ 36864 $ from $ 32736 $:

\[
32736 - 36864 = -4128
\]

### Step 3: Square root

We now have the expression:

\[
\sqrt{-4128}
\]

Since the square root of a negative number is imaginary, the solution becomes:

\[
\sqrt{-4128} = \sqrt{4128}i
\]

### Step 4: Simplify the square root of $ 4128 $

We can simplify $ \sqrt{4128} $:

\[
4128 = 64 \times 64.5
\]

So,

\[
\sqrt{4128} = 8\sqrt{64.5}i
\]

This is the simplified result, in terms of an imaginary number.

---

Would you like further explanation on any part of this process? Here are some related questions for you to explore:

1. How do you simplify square roots with imaginary numbers?
2. What is the meaning of the imaginary unit $ i $?
3. Can negative numbers inside square roots have real solutions?
4. How do you factor numbers like $ 4128 $ for simplification?
5. What are some real-world applications of imaginary numbers?

**Tip**: When dealing with negative square roots, always remember the solution involves imaginary numbers.

Learn how to solve the square root of the expression $$ \sqrt{8(4092) - 36864} $$, including steps for simplifying imaginary numbers. Understand key concepts like square roots, negative numbers, and imaginary solutions, suitable for high school students.

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