To solve this problem, we need to evaluate the expression:

$\frac{6.4 \times 10^5 + 1.9 \times 10^4}{5.7 \times 10^{-4}}$

Step-by-Step Solution

1. Evaluate the numerator: $6.4 \times 10^5 + 1.9 \times 10^4$ To add these terms, let's rewrite $1.9 \times 10^4$ as $0.19 \times 10^5$: $6.4 \times 10^5 + 0.19 \times 10^5 = (6.4 + 0.19) \times 10^5 = 6.59 \times 10^5$

2. Rewrite the denominator for easier calculation: The denominator is $5.7 \times 10^{-4}$.

3. Divide the results: Now, divide the numerator by the denominator: $\frac{6.59 \times 10^5}{5.7 \times 10^{-4}}$ Dividing these numbers involves dividing the coefficients and subtracting the exponents of 10: $= \frac{6.59}{5.7} \times 10^{5 - (-4)}$ Calculating $\frac{6.59}{5.7} \approx 1.1561$, and $5 - (-4) = 9$: $= 1.1561 \times 10^9$

4. Round to 2 significant figures: Rounding $1.1561$ to two significant figures gives us $1.2$. So the answer is: $1.2 \times 10^9$

Final Answer

$1.2 \times 10^9$

Would you like further details or have any questions?

Here are five related questions to expand on this topic:

1. How do you handle addition or subtraction of numbers in scientific notation with different exponents?
2. Why is it important to round to significant figures in scientific notation?
3. What are the general rules for multiplying and dividing numbers in scientific notation?
4. How does changing the exponent affect the scale of a number in scientific notation?
5. How can errors in rounding affect scientific calculations?

Tip: Always align the exponents in scientific notation before performing addition or subtraction.