Let us solve this step-by-step. The given mathematical problem is:

$\text{Calculate } \frac{3.4 \times 10^8 \times 8.2 \times 10^9}{5.7 \times 10^{-3} - 9.6 \times 10^{-4}} \text{ and express the answer in standard form to 2 significant figures.}$

Step 1: Simplify the denominator

The denominator is: $5.7 \times 10^{-3} - 9.6 \times 10^{-4}.$ Convert both terms to the same power of 10: $5.7 \times 10^{-3} = 57 \times 10^{-4}.$ So: $57 \times 10^{-4} - 9.6 \times 10^{-4} = (57 - 9.6) \times 10^{-4} = 47.4 \times 10^{-4}.$

Step 2: Simplify the numerator

The numerator is: $3.4 \times 10^8 \times 8.2 \times 10^9.$ Combine the numbers and powers of 10: $(3.4 \times 8.2) \times (10^8 \times 10^9) = 27.88 \times 10^{17}.$

Step 3: Combine numerator and denominator

Divide the simplified numerator by the simplified denominator: $\frac{27.88 \times 10^{17}}{47.4 \times 10^{-4}} = \frac{27.88}{47.4} \times 10^{17 - (-4)}.$ Simplify the numerical division and powers of 10: $\frac{27.88}{47.4} \approx 0.588 \quad \text{and} \quad 10^{17 - (-4)} = 10^{21}.$ Thus: $\frac{27.88 \times 10^{17}}{47.4 \times 10^{-4}} = 0.588 \times 10^{21}.$

Step 4: Convert to standard form and round to 2 significant figures

In standard form: $0.588 \times 10^{21} = 5.88 \times 10^{20}.$ Rounded to 2 significant figures: $\boxed{5.9 \times 10^{20}}.$

Do you want further details or have any questions?

Here are 5 related questions for deeper understanding:

1. How do you convert numbers with different powers of 10 for subtraction or addition?
2. Why is standard form important in scientific calculations?
3. What are some common mistakes when working with powers of 10?
4. How is significant figure rounding applied in different situations?
5. What are the rules for multiplying and dividing numbers in scientific notation?

Tip: Always check that numbers are expressed in consistent powers of 10 when performing addition or subtraction in scientific notation.