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Step 1:I'll solve this step by step using the precise LaTeX formatting guidelines you specified:
Step 2:: To find the decimal expansion of \frac{7}{22}, we need to perform long division.
Step 3:: Set up the division process
- Dividend: 7 - Divisor: 22
Step 4:: Perform long division
- First, multiply 7 by 10 to start the division: $$7 \times 10 = 70
- Divide 70 by 22: \frac{70}{22} = 3.18181818...
Step 5:: Analyze the decimal expansion
- The decimal repeats the pattern $$0.318181818...
- This means it's a repeating decimal with the digits 18 repeating infinitely
Step 6:: Verify the result
- $$\frac{7}{22} = 0.\overline{318}$$ (the bar indicates the repeating digits)
Final Answer
The decimal expansion of \frac{7}{22} is 0.318181818... or 0.\overline{318}, which is a repeating decimal where 18 repeats infinitely.
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