Part 1: Use the following numbers.
3.2 ✕ 104 1.95 ✕ 106 8.03 ✕ 10-8 2.56 ✕ 10-2 4.82 ✕ 1012
Choose two of the scientific numbers from the list and add them. Show your work.
Choose two of the scientific numbers from the list and subtract them. Show your work.
Choose two of the scientific numbers from the list and multiply them. Show your work.
Choose two of the scientific numbers from the list and divide them. Show your work.
Choose one of the scientific numbers from the list and add it to 32.8. Show your work.
Choose one of the scientific numbers from the list and multiply it by 256.7. Show your work.
- To add two scientific numbers, we add the numbers in front of the multiplication sign (coefficient) and keep the exponent the same. Let's choose 3.2 ✕ 10^4 and 1.95 ✕ 10^6.
3.2 ✕ 10^4 + 1.95 ✕ 10^6
= (3.2 + 1.95) ✕ 10^6
= 5.15 ✕ 10^6
- To subtract two scientific numbers, we subtract the numbers in front of the multiplication sign (coefficient) and keep the exponent the same. Let's choose 8.03 ✕ 10^-8 and 4.82 ✕ 10^12.
8.03 ✕ 10^-8 - 4.82 ✕ 10^12
= (8.03 - 4.82) ✕ 10^-8
= 3.21 ✕ 10^-8
- To multiply two scientific numbers, we multiply the numbers in front of the multiplication sign (coefficient) and add the exponents. Let's choose 3.2 ✕ 10^4 and 2.56 ✕ 10^-2.
(3.2 ✕ 10^4) * (2.56 ✕ 10^-2)
= (3.2 * 2.56) ✕ 10^(4 + -2)
= 8.192 ✕ 10^2
= 819.2
- To divide two scientific numbers, we divide the numbers in front of the multiplication sign (coefficient) and subtract the exponents. Let's choose 8.03 ✕ 10^-8 and 2.56 ✕ 10^-2.
(8.03 ✕ 10^-8) / (2.56 ✕ 10^-2)
= (8.03 / 2.56) ✕ 10^(-8 - -2)
= 3.13671875 ✕ 10^-6
≈ 3.14 ✕ 10^-6
- To add a scientific number and a decimal number, we keep the decimal number intact and adjust the exponent of the scientific number. Let's choose 3.2 ✕ 10^4 and add it to 32.8.
32.8 + (3.2 ✕ 10^4)
= 32.8 + 32,000
= 32032.8
- To multiply a scientific number by a decimal number, we keep the decimal number intact and add the exponents of the scientific number. Let's choose 1.95 ✕ 10^6 and multiply it by 256.7.
256.7 * (1.95 ✕ 10^6)
= 256.7 * 1.95 * 10^6
= 500.965 ✕ 10^6
= 5.00965 ✕ 10^8
To solve these equations, we need to understand how to perform arithmetic operations with scientific notation.
1. Adding two scientific numbers:
Let's choose 3.2 x 10^4 and 1.95 x 10^6.
To add these numbers, we need to make sure they have the same exponent. Let's convert 3.2 x 10^4 to 3.2 x 10^6.
3.2 x 10^4 + 1.95 x 10^6 = 3.2 x 10^6 + 1.95 x 10^6 = (3.2 + 1.95) x 10^6 = 5.15 x 10^6.
2. Subtracting two scientific numbers:
Let's choose 1.95 x 10^6 and 8.03 x 10^-8.
Again, we need to make the exponents the same. Converting 8.03 x 10^-8 to 8.03 x 10^6.
1.95 x 10^6 - (8.03 x 10^6) = (1.95 - 8.03) x 10^6 = -6.08 x 10^6.
3. Multiplying two scientific numbers:
Let's choose 8.03 x 10^-8 and 2.56 x 10^-2.
To multiply these numbers, we can simply multiply their coefficients and add their exponents.
(8.03 x 10^-8) * (2.56 x 10^-2) = (8.03 * 2.56) x (10^-8 * 10^-2) = 20.5568 x 10^-10 = 2.05568 x 10^-9.
4. Dividing two scientific numbers:
Let's choose 4.82 x 10^12 and 1.95 x 10^6.
To divide these numbers, we divide their coefficients and subtract their exponents.
(4.82 x 10^12) / (1.95 x 10^6) = (4.82 / 1.95) x (10^12 / 10^6) = 2.474358974 x 10^(12-6) = 2.474358974 x 10^6.
5. Adding a scientific number to a regular number:
Let's choose 1.95 x 10^6 and add it to 32.8.
We can simply add the regular number to the coefficient of the scientific notation.
1.95 x 10^6 + 32.8 = 1.95 x 10^6 + 32.8 = 1.95 x 10^6 + 3.28 x 10^1 = (1.95 + 3.28) x 10^6 = 5.23 x 10^6.
6. Multiplying a scientific number by a regular number:
Let's choose 3.2 x 10^4 and multiply it by 256.7.
We can simply multiply the regular number with the coefficient of the scientific notation.
3.2 x 10^4 * 256.7 = 3.2 * 256.7 x 10^4 = 822.4 x 10^4 = 8.224 x 10^5.
These are the steps to solve the given equations.
To perform the calculations, let's choose the numbers 3.2 ✕ 10^4 and 1.95 ✕ 10^6.
Addition:
3.2 ✕ 10^4 + 1.95 ✕ 10^6
= 3.2 ✕ 10^4 + 19.5 ✕ 10^5 (converting the second number to scientific notation)
= (3.2 + 19.5) ✕ 10^4 + 5 ✕ 10^5 (adding the numbers without changing the exponent)
= 22.7 ✕ 10^4 + 5 ✕ 10^5
= 2.27 ✕ 10^5 + 5 ✕ 10^5
= 7.27 ✕ 10^5 (combining the numbers with the same exponent)
Subtraction:
1.95 ✕ 10^6 - 3.2 ✕ 10^4
= 19.5 ✕ 10^5 - 0.32 ✕ 10^5 (converting the second number to scientific notation)
= (19.5 - 0.32) ✕ 10^5
= 19.18 ✕ 10^5
= 1.918 ✕ 10^6
Multiplication:
(3.2 ✕ 10^4) × (1.95 ✕ 10^6)
= (3.2 × 1.95) ✕ (10^4 × 10^6)
= 6.24 ✕ 10^10
Division:
(1.95 ✕ 10^6) ÷ (3.2 ✕ 10^4)
= (1.95 ÷ 3.2) ✕ (10^6 ÷ 10^4)
= 0.609375 ✕ 10^(6-4)
= 0.609375 ✕ 10^2
= 60.9375
Adding to 32.8:
3.2 ✕ 10^4 + 32.8
= 3.2 ✕ 10^4 + 32.8 ✕ 10^0 (converting the second number to scientific notation)
= (3.2 + 32.8) ✕ 10^4 + 0 ✕ 10^0 (adding the numbers without changing the exponent)
= 36 ✕ 10^4 + 0
= 36 ✕ 10^4
= 3.6 ✕ 10^5
Multiplying by 256.7:
(2.56 ✕ 10^-2) × 256.7
= 2.56 × 256.7 ✕ 10^-2 (ignoring the insignificant digits during calculation)
= 657.472 ✕ 10^-2
= 6.57472 ✕ 10^1 (converting back to scientific notation)