Let's work through each part of your request step by step.
Scientific Notation Calculations
1. Addition
Let's add \( 3.2 \times 10^4 \) and \( 1.95 \times 10^6 \).
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Convert \( 1.95 \times 10^6 \) to scientific notation with the same exponent as \( 3.2 \times 10^4 \): \[ 1.95 \times 10^6 = 1950000 \quad (Exponent reduced, effective conversion) \]
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Convert \( 3.2 \times 10^4 \) to match: \[ 3.2 \times 10^4 = 32000 \] Now we have: \[ 32000 + 1950000 = 1982000 \]
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Convert back to scientific notation: \[ 1982000 = 1.982 \times 10^6 \]
Final Answer: \( 3.2 \times 10^4 + 1.95 \times 10^6 = 1.982 \times 10^6 \)
2. Subtraction
We will subtract \( 2.56 \times 10^{-2} \) from \( 8.03 \times 10^{-8} \).
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First match the exponent: \[ 2.56 \times 10^{-2} = 256 \times 10^{-4} = 0.0000256 \] \[ 8.03 \times 10^{-8} = 0.0000000803 \]
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As \( 8.03 \times 10^{-8} < 2.56 \times 10^{-2} \), we write: \[ 8.03 \times 10^{-8} - 2.56 \times 10^{-2} = -2.55 \times 10^{-2} \]
Final Answer: \( 8.03 \times 10^{-8} - 2.56 \times 10^{-2} = -2.55 \times 10^{-2} \)
3. Multiplication
Multiply \( 4.82 \times 10^{12} \) and \( 2.56 \times 10^{-2} \).
\[ 4.82 \times 2.56 = 12.3392 \] Then add the exponents: \[ 10^{12} \cdot 10^{-2} = 10^{10} \] So the result: \[ = 12.3392 \times 10^{10} \] Convert to scientific notation: \[ = 1.23392 \times 10^{11} \]
Final Answer: \( 4.82 \times 10^{12} \times 2.56 \times 10^{-2} = 1.23392 \times 10^{11} \)
4. Division
Divide \( 1.95 \times 10^{6} \) by \( 8.03 \times 10^{-8} \).
\[ \frac{1.95}{8.03} \approx 0.2423 \] Now, subtract the exponents: \[ 10^{6} \div 10^{-8} = 10^{14} \] Thus, the result is: \[ = 0.2423 \times 10^{14} \] Convert to proper scientific notation: \[ = 2.423 \times 10^{13} \]
Final Answer: \( \frac{1.95 \times 10^{6}}{8.03 \times 10^{-8}} = 2.423 \times 10^{13} \)
5. Adding a number to a Scientific Number
Let's add \( 3.2 \times 10^{4} \) to \( 32.8 \).
Convert \( 32.8 \) to scientific notation: \[ 32.8 = 3.28 \times 10^{1} \] Now, convert \( 3.2 \times 10^{4} \): \[ 3.2 \times 10^{4} = 32000 \] Convert \( 3.28 \times 10^{1} \): \[ 3.28 \times 10^{1} = 32.8 \]
Adding: \[ 32000 + 32.8 = 32032.8 \]
Convert back to scientific notation: \[ = 3.20328 \times 10^{4} \]
Final Answer: \( 3.2 \times 10^4 + 32.8 = 3.20328 \times 10^4 \)
6. Subtracting from a number
Subtract \( 2.56 \times 10^{-2} \) from \( 1845.19 \).
Convert \( 2.56 \times 10^{-2} \): \[ = 0.0256 \] So, \[ 1845.19 - 0.0256 = 1845.1644 \]
Convert to scientific notation: \[ 1.8451644 \times 10^{3} \]
Final Answer: \( 1845.19 - 2.56 \times 10^{-2} = 1.8451644 \times 10^{3} \)
7. Multiplying by a number
Multiply \( 2.56 \times 10^{-2} \) by \( 256.7 \).
Calculate: \[ 2.56 \times 256.7 \approx 657.152 \]
Convert to scientific notation: \[ = 6.57152 \times 10^{2} \]
Final Answer: \( 2.56 \times 10^{-2} \times 256.7 = 6.57152 \times 10^{2} \)
8. Dividing by a number
Divide \( 4.82 \times 10^{12} \) by \( 0.3 \).
Calculate: \[ 4.82 \div 0.3 \approx 16.06667 \]
Convert to scientific notation: \[ = 1.606667 \times 10^{1} \times 10^{12} = 1.606667 \times 10^{13} \]
Final Answer: \( \frac{4.82 \times 10^{12}}{0.3} = 1.606667 \times 10^{13} \)
Travel Distance Problem
Plan a Trip:
- Atlanta to Baltimore, then Boston, then back to Atlanta.
- Distance from Atlanta to Baltimore: \( 9.2735 \times 10^{2} \)
- Distance from Baltimore to Boston: \( 5.7785 \times 10^{2} \)
- Distance from Boston back to Atlanta: \( 1.50511 \times 10^{3} \)
Total Distance Calculation: \[ Total = 9.2735 \times 10^{2} + 5.7785 \times 10^{2} + 1.50511 \times 10^{3} \] Converting to the same exponent: \[ = 9.2735 \times 10^{2} + 5.7785 \times 10^{2} + 15.0511 \times 10^{2} = 30.1031 \times 10^{2} \]
Convert to scientific notation: \[ = 3.01031 \times 10^{3} \]
Total Distance: \( \approx 3010.31 \) miles or \( 3.01031 \times 10^3 \) miles.
Total Distance for Sita's Trips
Distance to Seattle from Los Angeles: \( 3.50307 \times 10^{3} \).
Total Distance for three trips: \[ \text{Total} = 3 \times (3.50307 \times 10^{3}) = 1.050921 \times 10^{4} \]
Convert to scientific notation: \[ \text{Total Distance} = 1.050921 \times 10^{4} \]
Word Problem Using Division
Problem: If the distance between Los Angeles and Seattle is \( 3.50307 \times 10^{3} \) miles, how many trips can you take if you only have \( 8.5 \times 10^{3} \) miles worth of travel budget?
Solution: \[ \text{Trips} = \frac{8.5 \times 10^{3}}{3.50307 \times 10^{3}} \approx 2.43 \text{ trips} \]
Thus, you can take roughly 2 trips between Los Angeles and Seattle on that budget.
Planet Distances from Sun
(Here is an example of distances; if these are fictional characters, adjust as necessary):
- Mercury: \( 5.79 \times 10^{10} \)
- Venus: \( 1.08 \times 10^{11} \)
- Earth: \( 1.5 \times 10^{11} \)
- Mars: \( 2.28 \times 10^{11} \)
To add distances:
- Add Earth and Mars: \[ \text{Distance} = 1.5 \times 10^{11} + 2.28 \times 10^{11} = 3.78 \times 10^{11} \]
Final Answer: Distance of Earth + Mars = \( 3.78 \times 10^{11} \)
Comparative Analysis:
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Adding/Subtracting: To add/subtract numbers in scientific notation, you must align the exponents first; this often requires converting numbers such that both have the same powers of ten.
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Multiplying/Dividing: Multiplication and division involve straightforward arithmetic on the leading digits while adding or subtracting the exponents directly.
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Easiest Operation: Many find multiplying and dividing easier than adding or subtracting due to less conversion manipulation, but it depends on individual comfort with the math.
This comprehensive breakdown should help with all your requests. Let me know if you need further assistance!