Question

What is the step-by-step process to calculate the combined mass of two objects given in scientific notation, as described in the answer to the question about the mass of two skyscrapers?

Answer 1

To calculate the combined mass of two objects given in scientific notation, follow these steps:

1. **Identify the Masses**: Determine the masses of the two objects given in scientific notation. For example, let's say object 1 has a mass of \( m_1 = a_1 \times 10^{n_1} \) and object 2 has a mass of \( m_2 = a_2 \times 10^{n_2} \).

2. **Ensure Exponents are the Same**: If the exponents (\( n_1 \) and \( n_2 \)) are different, you need to adjust one of the masses so that both have the same power of ten. Choose the larger exponent for convenience. If \( n_1 < n_2 \), you can write:
\[
m_1 = a_1 \times 10^{n_1} = a_1 \times 10^{n_2 - (n_2 - n_1)} = a_1 \times 10^{n_2} \times 10^{(n_2 - n_1)}
\]
This means you will have:
\[
m_1 = (a_1 \times 10^{(n_2 - n_1)}) \times 10^{n_2}
\]

3. **Convert the First Mass**: Adjust \( m_1 \) (if needed) to match \( n_2 \):
\[
m_1' = a_1' \times 10^{n_2}
\]
where \( a_1' = a_1 \times 10^{(n_2 - n_1)} \).

4. **Combine the Coefficients**: Now, both masses are expressed with the same exponent:
\[
m_1' + m_2 = (a_1' + a_2) \times 10^{n_2}
\]

5. **Perform the Addition**: Add the coefficients \( a_1' \) and \( a_2 \):
\[
\text{Combine} = a_1' + a_2
\]

6. **Final Mass in Scientific Notation**: The result is:
\[
m_{total} = (\text{Combine}) \times 10^{n_2}
\]
If the result of the addition \( a_1' + a_2 \) exceeds 10, you would need to normalize it by adjusting the scientific notation. This means converting it to a form where the coefficient \( a \) is between 1 and 10.

7. **Convert if Necessary**: If \( a_{total} \) (the result of your addition) is 10 or greater:
\[
a_{total} = b \times 10^1 \quad \text{(where \( b < 10 \) )}
\]
Then, adjust the exponent accordingly:
\[
m_{total} = b \times 10^{(n_2 + 1)}
\]

8. **State the Result**: Present the combined mass in the final scientific notation.

### Example:
For instance, consider the masses \( m_1 = 2.5 \times 10^3 \, \text{kg} \) and \( m_2 = 3.1 \times 10^4 \, \text{kg} \):

1. The exponents: \( n_1 = 3 \), \( n_2 = 4 \) (common power \( 10^4 \)).
2. Convert \( m_1 \):
\[
m_1 = 2.5 \times 10^3 = 0.25 \times 10^4 \, \text{kg}
\]
3. Combine:
\[
0.25 \times 10^4 + 3.1 \times 10^4 = (0.25 + 3.1) \times 10^4 = 3.35 \times 10^4 \, \text{kg}
\]

This is your final result in scientific notation.