translate the number 362880 from base 10 to base 9 and explain how you did it

To convert the number \(362880\) from base 10 to base 9, we follow these steps:

1. **Identify the largest power of 9** less than or equal to \(362880\) to begin dividing:
   - Powers of 9: \(9^5 = 59049\), \(9^6 = 531441\), so we use \(9^5\) since \(9^6\) is too large.

2. **Divide and determine each coefficient** for each power of 9, starting from the highest power:
   - For \(9^5 = 59049\), divide \(362880\) by \(59049\):
     \[
     362880 \div 59049 = 6 \quad \text{(remainder: 59,586)}
     \]
     So, the coefficient for \(9^5\) is 6.
   - For \(9^4 = 6561\), divide \(59586\) by \(6561\):
     \[
     59586 \div 6561 = 9 \quad \text{(remainder: 5049)}
     \]
     So, the coefficient for \(9^4\) is 9.
   - For \(9^3 = 729\), divide \(5049\) by \(729\):
     \[
     5049 \div 729 = 6 \quad \text{(remainder: 423)}
     \]
     So, the coefficient for \(9^3\) is 6.
   - For \(9^2 = 81\), divide \(423\) by \(81\):
     \[
     423 \div 81 = 5 \quad \text{(remainder: 18)}
     \]
     So, the coefficient for \(9^2\) is 5.
   - For \(9^1 = 9\), divide \(18\) by \(9\):
     \[
     18 \div 9 = 2 \quad \text{(remainder: 0)}
     \]
     So, the coefficient for \(9^1\) is 2.
   - For \(9^0 = 1\), the remainder is \(0\), so the coefficient for \(9^0\) is 0.

3. **Combine the coefficients** from each power of 9 to form the final answer in base 9:
   - The result is \(696520_9\).

So, \(362880_{10}\) in base 9 is \(696520_9\).

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### Related Questions
1. How do we convert numbers from any base to another base?
2. What are other examples of converting numbers to base 9?
3. How would you verify the conversion from base 9 back to base 10?
4. How is the place value system used in base conversions?
5. Can we convert fractional numbers to base 9, and how?

### Tip
When converting between bases, always check for the highest power of the target base that is less than or equal to the number. This makes the process efficient and accurate.

Translate the number 362880 from base 10 to base 9 and explain how you did it.

Learn how to convert the decimal number 362880 to base 9 with a clear, detailed step-by-step guide. This solution uses the division-remainder method for base conversion, illustrating the coefficients for each power of 9. Perfect for students in Grades 8-10.

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