Question

In this problem, you will practice adding, subtracting, multiplying, and dividing numbers written in scientific notation. For each part, use the following numbers. 
  3.2×10^4       1.95×10^6        8.03×10^−8         2.56×10^−2         4.82×10^12

Answer 1

Sure! Let's perform operations using the given numbers in scientific notation:

1. **Addition:** \( 3.2 \times 10^4 + 1.95 \times 10^6 \)

To perform addition in scientific notation, we need to express both terms with the same exponent. It's easiest to rewrite \( 3.2 \times 10^4 \) with an exponent of \( 10^6 \):

\[
3.2 \times 10^4 = 0.032 \times 10^6
\]

Now we can add:

\[
0.032 \times 10^6 + 1.95 \times 10^6 = (0.032 + 1.95) \times 10^6 = 1.982 \times 10^6
\]

2. **Subtraction:** \( 1.95 \times 10^6 - 3.2 \times 10^4 \)

Using the same rewriting as above:

\[
1.95 \times 10^6 - 0.032 \times 10^6 = (1.95 - 0.032) \times 10^6 = 1.918 \times 10^6
\]

3. **Multiplication:** \( (3.2 \times 10^4) \times (4.82 \times 10^{12}) \)

To multiply numbers in scientific notation, multiply the coefficients and add the exponents:

\[
3.2 \times 4.82 = 15.424
\]
\[
10^4 \times 10^{12} = 10^{4 + 12} = 10^{16}
\]

Thus, the product is:

\[
15.424 \times 10^{16}
\]

We can convert this to proper scientific notation by rewriting \( 15.424 \) as \( 1.5424 \times 10^1 \):

\[
15.424 \times 10^{16} = 1.5424 \times 10^{17}
\]

4. **Division:** \( \frac{1.95 \times 10^6}{8.03 \times 10^{-8}} \)

To divide, divide the coefficients and subtract the exponents:

\[
\frac{1.95}{8.03} \approx 0.2433
\]
\[
10^{6 - (-8)} = 10^{6 + 8} = 10^{14}
\]

Therefore, the result is:

\[
0.2433 \times 10^{14}
\]

To convert this into proper scientific notation, we adjust \( 0.2433 \) to \( 2.433 \) and decrease the exponent by 1:

\[
2.433 \times 10^{13}
\]

In summary, the results are:
1. Addition: \( 1.982 \times 10^6 \)
2. Subtraction: \( 1.918 \times 10^6 \)
3. Multiplication: \( 1.5424 \times 10^{17} \)
4. Division: \( 2.433 \times 10^{13} \)