(9.70 ! 106) " (8.3 ! 105)
(9.70 ! 106) " (8.3 ! 105) # (9.70 ! 106) " (0.83 ! 106) # (9.70 " 0.83) ! 106
# 10.53 ! 106 # 1.053 ! 107

Let's carefully analyze the steps in your computation to see if the simplifications and the final result are correct.

You seem to be working with numbers expressed in scientific notation:

1. \((9.70 \times 10^6) \times (8.3 \times 10^5)\)
2. \((9.70 \times 10^6) \times (8.3 \times 10^5)\) is given as equivalent to \((9.70 \times 10^6) \times (0.83 \times 10^6)\).
3. The next step combines the numbers: \((9.70 \times 0.83) \times 10^6\).
4. This results in \(10.53 \times 10^6\).
5. Finally, \(10.53 \times 10^6\) is rewritten as \(1.053 \times 10^7\).

### Verification:

Let's go step by step to verify if everything is correct.

1. **Initial Expression**:
   \[
   (9.70 \times 10^6) \times (8.3 \times 10^5)
   \]

2. **Rewrite the multiplication**:
   \[
   (9.70 \times 8.3) \times (10^6 \times 10^5) = 80.51 \times 10^{11}
   \]

3. **Express \(80.51 \times 10^{11}\) in proper scientific notation**:
   \[
   8.051 \times 10^{12}
   \]

### Final Result:
The correct result is \(8.051 \times 10^{12}\), not \(1.053 \times 10^7\).

### Questions:
1. Why is \(10^{11}\) used in step 2 when multiplying \(10^6 \times 10^5\)?
2. Can the multiplication of \(9.70 \times 8.3\) be simplified further?
3. What happens if you multiply \(9.70 \times 0.83\) instead?
4. How does adjusting the decimal point affect the power of 10 in scientific notation?
5. How would this process change if you were multiplying powers with different bases?

### Tip:
Always ensure that the powers of 10 are handled correctly in each step to avoid errors in scientific notation.

Explore how to multiply numbers in scientific notation step-by-step. This problem involves numbers expressed in scientific notation and covers key concepts like multiplication rules in scientific notation. Suitable for high school students.

Math Problem Statement

10.53 ! 106 # 1.053 ! 107

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