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The properties of finding the arithmetic of numbers in scientific notation are given by:
- If the exponents of the scientific notation are the same, then the addition and subtraction can be determined easily.
- The multiplication and division of the scientific notation can be determined using the law of radicals.
Example:
1. Find the solution by adding \(6.83 \times 10^{20}\) and \(3.72 \times 10^{20}\)
Solution:
\(6.83 \times 10^{20} + 3.72 \times 10^{20} = (6.83 + 3.72) \times 10^{20} = 10.55 \times 10^{20}\)
Therefore, the solution is \(10.55 \times 10^{20}\).
2. Write \((6300000)^{2} \times (12000000)^3\) in scientific notation.
Solution:
\((600000)^{2} \times (2000000)^3 = (6 \times 10^5)^2 \times (2 \times 10^6)^3\)
\(= (6)^2 \times (10^5)^2 \times (2)^3 \times (10^6)^3\)
\(= (36) \times 10^{10} \times (8) \times 10^{18}\)
\(= (3.6 \times 10^1) \times 10^{10} \times (8) \times 10^{18}\)
\(= 3.6 \times 8 \times 10^1 \times 10^{10} \times 10^{18}\)
\(= 28.8 \times 10^{1+10+18}\)
\(= 2.88 \times 10^1 \times 10^{29}\)
\(= 2.88 \times 10^{30}\)
Therefore, the scientific notation is \(2.88 \times 10^{30}\).
3. Write \((200000000)^4 \div (0.00000004)^3\) in scientific notation.
Solution:
\((200000000)^6 \div (0.0000004)^3 = (2 \times 10^8)^6 \div (4 \times 10^{-7})^3\)
\(= \frac{(2)^6 \times (10^8)^6}{(4)^3 \times (10^{-7})^3}\)
\(= \frac{64 \times 10^{48}}{64 \times 10^{-21}}\)
\(= 1 \times 10^{48} \times 10^{21}\)
\(= 1 \times 10^{69}\)
Therefore, the solution is \(1 \times 10^{69}\).
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