Sure! Let's perform operations using the given numbers in scientific notation:
- 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:
\[ 0.032 \times 10^6 + 1.95 \times 10^6 = (0.032 + 1.95) \times 10^6 = 1.982 \times 10^6 \]
- Subtraction: \( 1.95 \times 10^6 - 3.2 \times 10^4 \)
Using the same rewriting as above:
\[ 1.95 \times 10^6 - 0.032 \times 10^6 = (1.95 - 0.032) \times 10^6 = 1.918 \times 10^6 \]
- 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 \( 1.5424 \times 10^1 \):
\[ 15.424 \times 10^{16} = 1.5424 \times 10^{17} \]
- 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:
- Addition: \( 1.982 \times 10^6 \)
- Subtraction: \( 1.918 \times 10^6 \)
- Multiplication: \( 1.5424 \times 10^{17} \)
- Division: \( 2.433 \times 10^{13} \)
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