Question
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
Answers
Answer
Simplify it and just show the work you used to come to this answer
Answer
What does /frac mean?
Answered by
GPT-4o mini
Sure! Let’s practice adding, subtracting, multiplying, and dividing the numbers \( 3.2 \times 10^4 \) and \( 1.95 \times 10^6 \).
### 1. Addition:
To add numbers in scientific notation, they must have the same exponent. We can rewrite \( 3.2 \times 10^4 \) to have the same exponent as \( 1.95 \times 10^6 \):
\[
3.2 \times 10^4 = 0.032 \times 10^6
\]
Now we can add them:
\[
0.032 \times 10^6 + 1.95 \times 10^6 = (0.032 + 1.95) \times 10^6 = 1.982 \times 10^6
\]
### 2. Subtraction:
We can use the same adjustment for subtraction:
\[
0.032 \times 10^6 - 1.95 \times 10^6 = (0.032 - 1.95) \times 10^6 = -1.918 \times 10^6
\]
### 3. Multiplication:
For multiplication, we multiply the coefficients and add the exponents:
\[
(3.2 \times 10^4) \times (1.95 \times 10^6) = (3.2 \times 1.95) \times (10^4 \times 10^6)
\]
\[
= 6.24 \times 10^{4 + 6} = 6.24 \times 10^{10}
\]
### 4. Division:
For division, we divide the coefficients and subtract the exponents:
\[
\frac{3.2 \times 10^4}{1.95 \times 10^6} = \left(\frac{3.2}{1.95}\right) \times \left(10^{4 - 6}\right)
\]
\[
= 1.641 \times 10^{-2} \quad (\text{approximately})
\]
### Summary of Results:
1. Addition: \( 1.982 \times 10^6 \)
2. Subtraction: \( -1.918 \times 10^6 \)
3. Multiplication: \( 6.24 \times 10^{10} \)
4. Division: \( 1.641 \times 10^{-2} \) (approximately)
### 1. Addition:
To add numbers in scientific notation, they must have the same exponent. We can rewrite \( 3.2 \times 10^4 \) to have the same exponent as \( 1.95 \times 10^6 \):
\[
3.2 \times 10^4 = 0.032 \times 10^6
\]
Now we can add them:
\[
0.032 \times 10^6 + 1.95 \times 10^6 = (0.032 + 1.95) \times 10^6 = 1.982 \times 10^6
\]
### 2. Subtraction:
We can use the same adjustment for subtraction:
\[
0.032 \times 10^6 - 1.95 \times 10^6 = (0.032 - 1.95) \times 10^6 = -1.918 \times 10^6
\]
### 3. Multiplication:
For multiplication, we multiply the coefficients and add the exponents:
\[
(3.2 \times 10^4) \times (1.95 \times 10^6) = (3.2 \times 1.95) \times (10^4 \times 10^6)
\]
\[
= 6.24 \times 10^{4 + 6} = 6.24 \times 10^{10}
\]
### 4. Division:
For division, we divide the coefficients and subtract the exponents:
\[
\frac{3.2 \times 10^4}{1.95 \times 10^6} = \left(\frac{3.2}{1.95}\right) \times \left(10^{4 - 6}\right)
\]
\[
= 1.641 \times 10^{-2} \quad (\text{approximately})
\]
### Summary of Results:
1. Addition: \( 1.982 \times 10^6 \)
2. Subtraction: \( -1.918 \times 10^6 \)
3. Multiplication: \( 6.24 \times 10^{10} \)
4. Division: \( 1.641 \times 10^{-2} \) (approximately)
Answered by
GPT-4o mini
Certainly! Here’s the simplified work for each operation, step by step:
### 1. Addition:
- Convert \( 3.2 \times 10^4 \) to the same exponent as \( 1.95 \times 10^6 \):
\[
3.2 \times 10^4 = 0.032 \times 10^6
\]
- Now 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:
- Using the same conversion:
\[
0.032 \times 10^6 - 1.95 \times 10^6 = (0.032 - 1.95) \times 10^6 = -1.918 \times 10^6
\]
### 3. Multiplication:
- Multiply the coefficients and add the exponents:
\[
(3.2 \times 10^4) \times (1.95 \times 10^6) = (3.2 \times 1.95) \times (10^{4 + 6})
\]
\[
= 6.24 \times 10^{10}
\]
### 4. Division:
- Divide the coefficients and subtract the exponents:
\[
\frac{3.2 \times 10^4}{1.95 \times 10^6} = \left(\frac{3.2}{1.95}\right) \times \left(10^{4 - 6}\right)
\]
\[
= 1.641 \times 10^{-2} \quad (\text{approximately})
\]
### Final Results:
1. **Addition**: \( 1.982 \times 10^6 \)
2. **Subtraction**: \( -1.918 \times 10^6 \)
3. **Multiplication**: \( 6.24 \times 10^{10} \)
4. **Division**: \( 1.641 \times 10^{-2} \) (approximately)
### 1. Addition:
- Convert \( 3.2 \times 10^4 \) to the same exponent as \( 1.95 \times 10^6 \):
\[
3.2 \times 10^4 = 0.032 \times 10^6
\]
- Now 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:
- Using the same conversion:
\[
0.032 \times 10^6 - 1.95 \times 10^6 = (0.032 - 1.95) \times 10^6 = -1.918 \times 10^6
\]
### 3. Multiplication:
- Multiply the coefficients and add the exponents:
\[
(3.2 \times 10^4) \times (1.95 \times 10^6) = (3.2 \times 1.95) \times (10^{4 + 6})
\]
\[
= 6.24 \times 10^{10}
\]
### 4. Division:
- Divide the coefficients and subtract the exponents:
\[
\frac{3.2 \times 10^4}{1.95 \times 10^6} = \left(\frac{3.2}{1.95}\right) \times \left(10^{4 - 6}\right)
\]
\[
= 1.641 \times 10^{-2} \quad (\text{approximately})
\]
### Final Results:
1. **Addition**: \( 1.982 \times 10^6 \)
2. **Subtraction**: \( -1.918 \times 10^6 \)
3. **Multiplication**: \( 6.24 \times 10^{10} \)
4. **Division**: \( 1.641 \times 10^{-2} \) (approximately)
Answered by
GPT-4o mini
The command `/frac` is a common notation used in typesetting to create a fraction. In LaTeX, which is a typesetting system often used for mathematical and scientific documents, the command `\frac{numerator}{denominator}` is used to display fractions in a clear format.
For example:
- If you want to represent the fraction \( \frac{a}{b} \), you would write `\frac{a}{b}` in LaTeX.
In plain text, it usually just means to denote division or a fractional relationship, such as writing \( \frac{3.2}{1.95} \) to indicate that 3.2 is divided by 1.95.
If you're asking about a specific context where you've seen `/frac`, please let me know, and I can provide more tailored information!
For example:
- If you want to represent the fraction \( \frac{a}{b} \), you would write `\frac{a}{b}` in LaTeX.
In plain text, it usually just means to denote division or a fractional relationship, such as writing \( \frac{3.2}{1.95} \) to indicate that 3.2 is divided by 1.95.
If you're asking about a specific context where you've seen `/frac`, please let me know, and I can provide more tailored information!