The problem in the image asks to simplify the given expression:

$\frac{6.14 \times 10^4}{(5.60 \times 10^{-2} \times 2.408 \times 10^0)}$

into the form $b \times 10^n$, where $b$ is correct to two decimal places, and $n$ is an integer.

Let's break it down:

Step 1: Simplify the powers of 10

For the denominator: $5.60 \times 10^{-2} \times 2.408 \times 10^0 = (5.60 \times 2.408) \times 10^{-2}$ Calculate the product: $5.60 \times 2.408 = 13.4848$ Thus, the denominator becomes: $13.4848 \times 10^{-2}$

Step 2: Write the expression with powers of 10 separated

Now the expression becomes: $\frac{6.14 \times 10^4}{13.4848 \times 10^{-2}} = \frac{6.14}{13.4848} \times 10^{4 - (-2)} = \frac{6.14}{13.4848} \times 10^6$

Step 3: Simplify the fraction

Calculate the fraction: $\frac{6.14}{13.4848} \approx 0.4555$

Step 4: Express in standard form

Now the expression is: $0.4555 \times 10^6$

To express this in standard scientific notation, adjust the decimal to make the coefficient $b$ between 1 and 10: $0.4555 \times 10^6 = 4.555 \times 10^5$

Step 5: Round to two decimal places

Rounding $b$ to two decimal places gives: $b = 4.56, \quad n = 5$

Thus, the final answer is: $b = 4.56, \quad n = 5$

Let me know if you'd like more details or further clarifications! Here are five related questions you might want to explore:

1. How do you simplify an expression with multiple powers of 10?
2. What is scientific notation and why is it useful in these problems?
3. How do you multiply and divide numbers in scientific notation?
4. Why do you need to adjust the decimal to get the final $b$-value between 1 and 10?
5. What are common mistakes to avoid when simplifying expressions with powers of 10?

Tip: When dealing with scientific notation, always focus on separating the decimal and power of 10 parts first—it simplifies the process!