Theory:

The properties of finding the arithmetic of numbers in scientific notation are given by:
  1. If the exponents of the scientific notation are the same, then the addition and subtraction can be determined easily.
  2. 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 \(1 \times 10^{69}\).