Let's solve each part step by step.

Part (a)

The problem is: $4,000,000 \cdot 200 = a \cdot 10^m$

First, express each number in scientific notation:

• $4,000,000 = 4 \times 10^6$
• $200 = 2 \times 10^2$

Now, multiply these two numbers: $(4 \times 10^6) \cdot (2 \times 10^2) = (4 \cdot 2) \times (10^6 \cdot 10^2) = 8 \times 10^{6 + 2} = 8 \times 10^8$ Therefore, $a = 8$ and $m = 8$.

Part (b)

The problem is: $1.6 \text{ million} \div 800 = b \cdot 10^n$

First, express each number in scientific notation:

• $1.6 \text{ million} = 1.6 \times 10^6$
• $800 = 8 \times 10^2$

Now, divide these two numbers: $\frac{1.6 \times 10^6}{8 \times 10^2} = \left(\frac{1.6}{8}\right) \times 10^{6 - 2} = 0.2 \times 10^4$ To express $0.2 \times 10^4$ in scientific notation, convert it to $2 \times 10^{-1} \times 10^4 = 2 \times 10^{3}$.

Thus, $b = 2$ and $n = 3$.

Final Answers

• For (a): $a = 8$, $m = 8$
• For (b): $b = 2$, $n = 3$

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you convert large numbers into scientific notation?
2. What is the process for multiplying numbers in scientific notation?
3. How do you divide numbers in scientific notation?
4. Why is it important to adjust scientific notation to have one digit to the left of the decimal point?
5. How do exponents add or subtract when multiplying or dividing in scientific notation?

Tip: When working with scientific notation, always ensure that your result has a single non-zero digit to the left of the decimal point for proper formatting.