A:Make a list of the distances from each planet to the sun, written in scientific notation. If there is a shortest and longest distance, use the average distance from the sun.
B:Choose two of the distances of planets from the sun and add them. Show your work.
C:Choose two of the distances of planets from the sun and subtract them. Show your work.
D:Choose two of the distances of planets from the sun and multiply them. Show your work.
E:Choose two of the distances of planets from the sun and divide them. Show your work
4:Compare and contrast adding/subtracting numbers written in scientific notation with multiplying/dividing numbers written in scientific notation. How are the steps in each case similar? How are they different? Do you think adding, subtracting, multiplying, or dividing numbers written in scientific notation is easiest? Why?
A: Distances from planets to the sun in scientific notation:
- Mercury: 5.79 x 10^7 km
- Venus: 1.08 x 10^8 km
- Earth: 1.50 x 10^8 km
- Mars: 2.28 x 10^8 km
- Jupiter: 7.78 x 10^8 km
- Saturn: 1.43 x 10^9 km
- Uranus: 2.87 x 10^9 km
- Neptune: 4.50 x 10^9 km
Since there is no shortest and longest distance in this case, we can simply use the average distance from the sun, which is approximately 2.87 x 10^9 km.
B: Choosing distances from Mars and Neptune:
Distance from Mars: 2.28 x 10^8 km
Distance from Neptune: 4.50 x 10^9 km
Adding the distances: (2.28 x 10^8 km) + (4.50 x 10^9 km) = 4.73 x 10^9 km
C: Choosing distances from Venus and Mercury:
Distance from Venus: 1.08 x 10^8 km
Distance from Mercury: 5.79 x 10^7 km
Subtracting the distances: (1.08 x 10^8 km) - (5.79 x 10^7 km) = 5.0 x 10^7 km
D: Choosing distances from Earth and Saturn:
Distance from Earth: 1.50 x 10^8 km
Distance from Saturn: 1.43 x 10^9 km
Multiplying the distances: (1.50 x 10^8 km) * (1.43 x 10^9 km) = 2.15 x 10^17 km^2
E: Choosing distances from Jupiter and Uranus:
Distance from Jupiter: 7.78 x 10^8 km
Distance from Uranus: 2.87 x 10^9 km
Dividing the distances: (7.78 x 10^8 km) / (2.87 x 10^9 km) = 0.27
4: When adding or subtracting numbers written in scientific notation, we need to ensure that the exponents are the same. We can then perform the addition or subtraction on the coefficients while keeping the same exponent. When multiplying numbers in scientific notation, we multiply the coefficients and add the exponents. When dividing numbers in scientific notation, we divide the coefficients and subtract the exponents.
The steps involved in each case are similar as we need to rearrange the numbers to ensure the exponents are the same and then perform the operation on the coefficients. However, the operations and the ways in which we manipulate the exponents differ.
Adding or subtracting numbers written in scientific notation is generally easier as we only need to adjust the exponents and perform addition or subtraction on the coefficients. Multiplying or dividing numbers written in scientific notation involves both multiplication or division of the coefficients and manipulation of the exponents, which can be more complex.
A: List of distances from each planet to the sun written in scientific notation:
1. Mercury: 5.79 x 10^7 km
2. Venus: 1.08 x 10^8 km
3. Earth: 1.50 x 10^8 km
4. Mars: 2.28 x 10^8 km
5. Jupiter: 7.78 x 10^8 km
6. Saturn: 1.43 x 10^9 km
7. Uranus: 2.87 x 10^9 km
8. Neptune: 4.50 x 10^9 km
To find the average distance, add all the distances and divide by 8.
Average distance = (5.79 x 10^7 + 1.08 x 10^8 + 1.50 x 10^8 + 2.28 x 10^8 + 7.78 x 10^8 + 1.43 x 10^9 + 2.87 x 10^9 + 4.50 x 10^9) / 8
B: Adding two distances:
Let's choose Earth and Jupiter.
Earth's distance: 1.50 x 10^8 km
Jupiter's distance: 7.78 x 10^8 km
Adding the distances:
1.50 x 10^8 + 7.78 x 10^8 = (1.50 + 7.78) x 10^8 = 9.28 x 10^8 km
C: Subtracting two distances:
Let's choose Mars and Venus.
Mars's distance: 2.28 x 10^8 km
Venus's distance: 1.08 x 10^8 km
Subtracting the distances:
2.28 x 10^8 - 1.08 x 10^8 = (2.28 - 1.08) x 10^8 = 1.20 x 10^8 km
D: Multiplying two distances:
Let's choose Jupiter and Uranus.
Jupiter's distance: 7.78 x 10^8 km
Uranus's distance: 2.87 x 10^9 km
Multiplying the distances:
7.78 x 10^8 * 2.87 x 10^9 = (7.78 * 2.87) x (10^8 * 10^9) = 22.367 x 10^17 km
E: Dividing two distances:
Let's choose Saturn and Mercury.
Saturn's distance: 1.43 x 10^9 km
Mercury's distance: 5.79 x 10^7 km
Dividing the distances:
1.43 x 10^9 / 5.79 x 10^7 = (1.43 / 5.79) x (10^9 / 10^7) = 0.247 x 10^2 = 24.7
4: When adding or subtracting numbers in scientific notation, the step is similar to regular addition or subtraction, where the exponents are kept the same, and the coefficients are added or subtracted. When multiplying numbers in scientific notation, the exponents are added, and the coefficients are multiplied. When dividing numbers in scientific notation, the exponents are subtracted, and the coefficients are divided.
The steps are similar in that the exponents are manipulated based on the operation and the coefficients are operated upon. However, the main difference lies in the specific arithmetic operation being performed (addition/subtraction vs. multiplication/division) and the operation applied to the coefficients.
In terms of ease, it may vary from person to person. Generally, adding and subtracting numbers in scientific notation is relatively straightforward, as it involves manipulating the coefficients while keeping the exponents unchanged. Multiplying and dividing numbers in scientific notation require additional operations with the exponents, but both can be deemed comparatively manageable with practice.
A: To make a list of distances from each planet to the sun in scientific notation, you need to know the average distances of the planets from the sun. Here is an example list:
1. Mercury: 5.79 x 10^7 km
2. Venus: 1.08 x 10^8 km
3. Earth: 1.50 x 10^8 km
4. Mars: 2.28 x 10^8 km
5. Jupiter: 7.78 x 10^8 km
6. Saturn: 1.43 x 10^9 km
7. Uranus: 2.87 x 10^9 km
8. Neptune: 4.50 x 10^9 km
If there were a shortest and longest distance, you would use the average distance from the sun instead.
B: To add two distances of planets from the sun, you simply add the numbers written in scientific notation. For example, if we choose Earth and Mars:
1.50 x 10^8 km + 2.28 x 10^8 km = 3.78 x 10^8 km
You add the numbers in front and keep the same exponent in scientific notation.
C: To subtract two distances of planets from the sun, you subtract the numbers written in scientific notation. For example, if we choose Venus and Mars:
1.08 x 10^8 km - 2.28 x 10^8 km = -1.20 x 10^8 km
Again, you subtract the numbers in front and keep the same exponent in scientific notation.
D: To multiply two distances of planets from the sun, you multiply the numbers written in scientific notation. For example, if we choose Earth and Jupiter:
1.50 x 10^8 km * 7.78 x 10^8 km = 1.17 x 10^17 km^2
You multiply the numbers in front and add the exponents in scientific notation. In this case, the exponents are added because we are multiplying.
E: To divide two distances of planets from the sun, you divide the numbers written in scientific notation. For example, if we choose Saturn and Neptune:
1.43 x 10^9 km / 4.50 x 10^9 km = 0.32 x 10^0 km
You divide the numbers in front and subtract the exponents in scientific notation. In this case, the exponents are subtracted because we are dividing.
4: Adding and subtracting numbers written in scientific notation is similar because you focus on adding or subtracting the numbers in front and keeping the same exponent. The steps involve manipulating the numbers without changing their relative magnitudes, just like with regular arithmetic.
On the other hand, multiplying and dividing numbers written in scientific notation involve multiplying or dividing the numbers in front and adding or subtracting the exponents. The steps differ from adding and subtracting as you need to perform additional operations with the exponents.
The easiest operation among adding, subtracting, multiplying, or dividing numbers written in scientific notation may vary for different individuals. Some find adding and subtracting simpler because the exponent manipulation is not involved. Others may find multiplying and dividing easier due to the straightforward pattern of adding or subtracting exponents. It ultimately depends on an individual's familiarity and comfort with scientific notation and basic arithmetic operations.