To solve this problem, we need to simplify the given mathematical expressions and then compare the results to arrange them from least to greatest. Let's break down each of the provided expressions.

1. First Expression: $(4.27 \times 10^8) \cdot (9.2 \times 10^{-5})$ To multiply, we combine the constants and powers of 10: $4.27 \times 9.2 = 39.284$ For the powers of 10: $10^8 \times 10^{-5} = 10^{8 + (-5)} = 10^3$ So the result is: $39.284 \times 10^3 = 3.9284 \times 10^4$

2. Second Expression: $(6.96 \times 10^5) \cdot (8.43 \times 10^{-2})$ Multiplying the constants: $6.96 \times 8.43 = 58.6728$ For the powers of 10: $10^5 \times 10^{-2} = 10^{5 - 2} = 10^3$ So the result is: $58.6728 \times 10^3 = 5.86728 \times 10^4$

3. Third Expression: $\frac{(7.25 \times 10^6)}{(9.36 \times 10^{-7})}$ Dividing the constants: $7.25 \div 9.36 = 0.77479$ For the powers of 10: $10^6 \div 10^{-7} = 10^{6 - (-7)} = 10^{13}$ So the result is: $0.77479 \times 10^{13} = 7.7479 \times 10^{12}$

4. Fourth Expression: $\frac{(8.32 \times 10^4)}{(9.67 \times 10^1)}$ Dividing the constants: $8.32 \div 9.67 = 0.8607$ For the powers of 10: $10^4 \div 10^1 = 10^{4 - 1} = 10^3$ So the result is: $0.8607 \times 10^3 = 8.607 \times 10^2$

5. Fifth Expression: $\frac{(1.89 \times 10^4)}{(5.92 \times 10^3)}$ Dividing the constants: $1.89 \div 5.92 = 0.31926$ For the powers of 10: $10^4 \div 10^3 = 10^{4 - 3} = 10^1$ So the result is: $0.31926 \times 10^1 = 3.1926$

Now, let’s arrange the results from least to greatest:

1. $3.1926$
2. $8.607 \times 10^2$
3. $3.9284 \times 10^4$
4. $5.86728 \times 10^4$
5. $7.7479 \times 10^{12}$

Would you like further clarification on any of the steps? Here are 5 related questions you might want to explore:

1. How can powers of 10 be manipulated when dividing numbers?
2. What are some tips for simplifying multiplication with scientific notation?
3. How do you approach division problems involving very large or small numbers in scientific notation?
4. What is the difference between multiplying and dividing exponents?
5. Can scientific notation be used for comparing very large and very small values effectively?

Tip: When multiplying or dividing numbers in scientific notation, handle the constants and powers of 10 separately, then combine them at the end.