Certainly! Let's dive deeper into the third step of the calculation:

### Step 3: Division Involving Scientific Notation
The expression to evaluate is:

\[
\frac{1}{0.166 \cdot 10^{-23}}
\]

This involves dividing 1 by the product of \(0.166\) and \(10^{-23}\). We can break this down as follows:

- **Scientific Notation:** \(10^{-23}\) is scientific notation, representing a very small number: \(0.00000000000000000000001\).

- **Multiplication:** When multiplying a number by \(10^{-23}\), you are effectively shifting the decimal point 23 places to the left.

- **Division:** To divide by a product like \(0.166 \cdot 10^{-23}\), you can separate the operation into two parts:
  - First, divide by \(0.166\).
  - Second, multiply by the inverse of \(10^{-23}\), which is \(10^{23}\).

So the division:

\[
\frac{1}{0.166 \cdot 10^{-23}}
\]

can be rewritten as:

\[
\frac{1}{0.166} \cdot \frac{1}{10^{-23}}
\]

This simplifies to:

\[
\frac{1}{0.166} \times 10^{23}
\]

### Why This Works:
- **Dividing by a small number (like \(0.166\)) increases the value** because you’re splitting 1 into smaller pieces, which results in a larger number.
- **Multiplying by \(10^{23}\) shifts the decimal point** 23 places to the right, turning a small value into a very large one.

So, the result after dividing \(1\) by \(0.166 \times 10^{-23}\) becomes:

\[
6.0241 \times 10^{23}
\]

Let me know if this clears it up, or if you have any further questions!

---

Here are some follow-up questions that might help:
1. What does \(10^{-23}\) represent in terms of magnitude?
2. How do you handle operations with numbers in scientific notation?
3. What is the effect of multiplying or dividing by numbers like \(0.166\)?
4. How does scientific notation simplify the handling of very large or small numbers?
5. What are other examples where scientific notation is essential?

**Tip:** When performing calculations with very large or very small numbers, breaking the problem into smaller, manageable steps, as done here, can help ensure accuracy and clarity.

Explore the division involving scientific notation step-by-step. Learn how to divide numbers in scientific notation, handling small and large values effectively. Suitable for advanced high school or college-level students.

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