Question
Find
find\frac{\left(3.2\cdot10^{10}\right)\left(3.75\cdot10^{6}\right)}{\left(4\cdot10^{6}\right)\left(6\cdot10^{5}\right)} , expressed in scientific notation.
Responses
find\frac{\left(3.2\cdot10^{10}\right)\left(3.75\cdot10^{6}\right)}{\left(4\cdot10^{6}\right)\left(6\cdot10^{5}\right)} , expressed in scientific notation.
Responses
Answers
GPT-4o mini
To simplify the expression
\[
\frac{(3.2 \cdot 10^{10})(3.75 \cdot 10^{6})}{(4 \cdot 10^{6})(6 \cdot 10^{5})},
\]
we can first simplify the coefficients and the powers of ten separately.
### Step 1: Simplify the coefficients
First, calculate the numerator coefficients:
\[
3.2 \cdot 3.75.
\]
Calculating this:
\[
3.2 \cdot 3.75 = 12.0.
\]
Now calculate the denominator coefficients:
\[
4 \cdot 6 = 24.
\]
### Step 2: Combine the powers of ten
Next, for the powers of ten in the numerator and denominator:
In the numerator:
\[
10^{10} \cdot 10^{6} = 10^{10+6} = 10^{16}.
\]
In the denominator:
\[
10^{6} \cdot 10^{5} = 10^{6+5} = 10^{11}.
\]
### Step 3: Create the new fraction
Now we combine the results:
The overall expression becomes:
\[
\frac{12.0 \cdot 10^{16}}{24 \cdot 10^{11}}.
\]
This can be split into:
\[
\frac{12.0}{24} \cdot \frac{10^{16}}{10^{11}}.
\]
Calculating the fraction of the coefficients:
\[
\frac{12.0}{24} = 0.5.
\]
And for the powers of ten:
\[
10^{16-11} = 10^{5}.
\]
### Step 4: Combine into scientific notation
Now we combine them:
\[
0.5 \cdot 10^{5}.
\]
To express \(0.5\) in proper scientific notation, we can say:
\[
0.5 = 5.0 \cdot 10^{-1}.
\]
Thus, we can write the expression as:
\[
5.0 \cdot 10^{-1} \cdot 10^{5} = 5.0 \cdot 10^{4}.
\]
### Final Answer
Therefore, the final result in scientific notation is:
\[
\boxed{5.0 \cdot 10^{4}}.
\]
\[
\frac{(3.2 \cdot 10^{10})(3.75 \cdot 10^{6})}{(4 \cdot 10^{6})(6 \cdot 10^{5})},
\]
we can first simplify the coefficients and the powers of ten separately.
### Step 1: Simplify the coefficients
First, calculate the numerator coefficients:
\[
3.2 \cdot 3.75.
\]
Calculating this:
\[
3.2 \cdot 3.75 = 12.0.
\]
Now calculate the denominator coefficients:
\[
4 \cdot 6 = 24.
\]
### Step 2: Combine the powers of ten
Next, for the powers of ten in the numerator and denominator:
In the numerator:
\[
10^{10} \cdot 10^{6} = 10^{10+6} = 10^{16}.
\]
In the denominator:
\[
10^{6} \cdot 10^{5} = 10^{6+5} = 10^{11}.
\]
### Step 3: Create the new fraction
Now we combine the results:
The overall expression becomes:
\[
\frac{12.0 \cdot 10^{16}}{24 \cdot 10^{11}}.
\]
This can be split into:
\[
\frac{12.0}{24} \cdot \frac{10^{16}}{10^{11}}.
\]
Calculating the fraction of the coefficients:
\[
\frac{12.0}{24} = 0.5.
\]
And for the powers of ten:
\[
10^{16-11} = 10^{5}.
\]
### Step 4: Combine into scientific notation
Now we combine them:
\[
0.5 \cdot 10^{5}.
\]
To express \(0.5\) in proper scientific notation, we can say:
\[
0.5 = 5.0 \cdot 10^{-1}.
\]
Thus, we can write the expression as:
\[
5.0 \cdot 10^{-1} \cdot 10^{5} = 5.0 \cdot 10^{4}.
\]
### Final Answer
Therefore, the final result in scientific notation is:
\[
\boxed{5.0 \cdot 10^{4}}.
\]