Sure! Let's carefully walk through each part of the calculation step-by-step.
### Step 1: Understand the Expression
We need to perform the operation:
[tex]\[ \frac{9 \times 10^2}{3 \times 10^{12}} \][/tex]
### Step 2: Simplify the Coefficient
First, simplify the numerical coefficients separately from the powers of 10:
[tex]\[ \frac{9}{3} = 3 \][/tex]
### Step 3: Apply the Laws of Exponents
Next, apply the laws of exponents to the powers of 10. In general, for [tex]\(\frac{a \times 10^m}{b \times 10^n}\)[/tex]:
[tex]\[ \frac{a \times 10^m}{b \times 10^n} = \left( \frac{a}{b} \right) \times 10^{m-n} \][/tex]
So, we subtract the exponents:
[tex]\[ 10^2 \div 10^{12} = 10^{2-12} = 10^{-10} \][/tex]
### Step 4: Combine the Results
From the simplifications above, we now have:
[tex]\[ \frac{9 \times 10^2}{3 \times 10^{12}} = 3 \times 10^{-10} \][/tex]
### Step 5: Express the Answer in Correct Scientific Notation
The result is already in proper scientific notation. Recall that scientific notation is of the form:
[tex]\[ a \times 10^b \][/tex]
where [tex]\(1 \leq |a| < 10\)[/tex] and [tex]\(b\)[/tex] is an integer. Here:
[tex]\[ a = 3.0 \][/tex]
[tex]\[ b = -10 \][/tex]
Thus, the result is:
[tex]\[ 3.0 \times 10^{-10} \][/tex]
In conclusion:
[tex]\[ \frac{9 \times 10^2}{3 \times 10^{12}} = 3.0 \times 10^{-10} \][/tex]
