Here are the calculations for the portfolio activities involving scientific notation:
Addition
Choose two scientific numbers:
3.2×10^4 and 1.95×10^6.
Add them: To add these numbers, we need to express them with the same exponent. We choose to express both with the exponent of 10^6.
- Convert 3.2×10^4 to the same exponent: \[ 3.2 \times 10^4 = 0.032 \times 10^6 \]
Now add: \[ 0.032 \times 10^6 + 1.95 \times 10^6 = (0.032 + 1.95) \times 10^6 = 1.982 \times 10^6 \]
Subtraction
Choose two scientific numbers:
2.56×10^−2 and 8.03×10^−8.
Subtract them: We will express both numbers with the same exponent. We can convert 2.56×10⁻² to an exponent of 10⁻⁸:
- Convert 2.56×10⁻²: \[ 2.56 \times 10^{-2} = 2.56 \times 10^{-2 + 6} = 2.56 \times 10^{-2} = 256 \times 10^{-8} \]
Now subtract: \[ 256 \times 10^{-8} - 8.03 \times 10^{-8} = (256 - 8.03) \times 10^{-8} = 247.97 \times 10^{-8} \] which can be simplified to: \[ 2.4797 \times 10^{-6} \]
Multiplication
Choose two scientific numbers:
4.82×10^12 and 8.03×10^−8.
Multiply them: \[ 4.82 \times 10^{12} \times 8.03 \times 10^{-8} = (4.82 \times 8.03) \times (10^{12} \times 10^{-8}) \]
Calculating: \[ 4.82 \times 8.03 = 38.7056 \] \[ 10^{12} \times 10^{-8} = 10^{12-8} = 10^4 \]
Combining results: \[ 38.7056 \times 10^4 = 3.87056 \times 10^5 \]
Division
Choose two scientific numbers:
1.95×10^6 divided by 3.2×10^4.
Divide them: \[ \frac{1.95 \times 10^{6}}{3.2 \times 10^{4}} = \frac{1.95}{3.2} \times 10^{6 - 4} \]
Calculating: \[ \frac{1.95}{3.2} \approx 0.609375 \] \[ 10^{6 - 4} = 10^2 \]
Combining results: \[ 0.609375 \times 10^2 = 6.09375 \times 10^1 \]
Adding Scientific Number to a Decimal
Choose a scientific number:
1.95×10^6 and add it to 32.8.
Add: First, convert 32.8 to scientific notation: \[ 32.8 = 3.28 \times 10^1 \] Now express both in terms of a similar exponent (10^6): \[ 3.28 \times 10^1 = 0.0000328 \times 10^6 \]
Now add: \[ 1.95 \times 10^6 + 0.0000328 \times 10^6 = (1.95 + 0.0000328) \times 10^6 = 1.9500328 \times 10^6 \]
Subtracting from a Decimal
Choose a scientific number:
2.56×10^−2 and subtract it from 1,845.19.
Subtract: First, convert 1,845.19 to scientific notation: \[ 1,845.19 = 1.84519 \times 10^3 \]
Now convert 2.56×10^−2 to the same exponent (10³): \[ 2.56 \times 10^{-2} = 0.0000256 \times 10^3 \]
Now perform the subtraction: \[ 1.84519 \times 10^3 - 0.0000256 \times 10^3 = (1.84519 - 0.0000256) \times 10^3 = 1.8451644 \times 10^3 \]
Multiplying by a Decimal
Choose a scientific number:
2.56×10^−2 and multiply it by 256.7.
Multiply: \[ 2.56 \times 10^{-2} \times 256.7 \]
Calculating: \[ 2.56 \times 256.7 = 657.472 \]
Now express in scientific notation: \[ 657.472 = 6.57472 \times 10^2 \]
Final result: \[ 6.57472 \times 10^2 \times 10^{-2} = 6.57472 \times 10^0 = 6.57472 \]
Dividing by a Decimal
Choose a scientific number:
8.03×10^−8 and divide it by 0.3.
Divide: \[ \frac{8.03 \times 10^{-8}}{0.3} = 8.03 \times \frac{10^{-8}}{0.3} \]
Calculating: \[ \frac{8.03}{0.3} = 26.7667 \] This would also be written as: \[ 26.7667 \times 10^{-8} = 2.67667 \times 10^{-7} \]
Summary of Results
- Addition: \( 1.982 \times 10^6 \)
- Subtraction: \( 2.4797 \times 10^{-6} \)
- Multiplication: \( 3.87056 \times 10^5 \)
- Division: \( 6.09375 \times 10^1 \)
- Add to 32.8: \( 1.9500328 \times 10^6 \)
- Subtract from 1,845.19: \( 1.8451644 \times 10^3 \)
- Multiply by 256.7: \( 6.57472 \)
- Divide by 0.3: \( 2.67667 \times 10^{-7} \)
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