To solve the expression \(\frac{1 \cdot 1}{0.166 \cdot 10^{-23}}\):

1. Simplify the numerator:
   \[
   1 \cdot 1 = 1
   \]
   
2. Simplify the denominator:
   \[
   0.166 \cdot 10^{-23}
   \]

3. Now, perform the division:
   \[
   \frac{1}{0.166 \cdot 10^{-23}} = \frac{1}{0.166} \cdot 10^{23} 
   \]

4. Calculate \(\frac{1}{0.166}\):
   \[
   \frac{1}{0.166} \approx 6.0241
   \]

5. Multiply by \(10^{23}\):
   \[
   6.0241 \times 10^{23} = 6.0241 \times 10^{23}
   \]

So, the result is approximately:
\[
6.0241 \times 10^{23}
\]

Would you like more details or have any questions?

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Here are some related questions you might find interesting:
1. What is the significance of scientific notation in mathematics?
2. How do you convert between standard form and scientific notation?
3. What are some common applications of numbers in the form \(a \times 10^n\)?
4. How can you simplify expressions with negative exponents?
5. How does one estimate the value of a division involving very small numbers?

**Tip:** When dealing with very small or very large numbers, scientific notation helps simplify calculations and improves clarity.

Explore how to solve division involving very small numbers using scientific notation. This problem involves dividing 1 by 0.166 times 10 to the power of -23, demonstrating techniques for handling small values in calculations.

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