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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}.