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The image shows a mathematical problem that involves converting the number \( (523)_6 \) from base 6 to base 9.

To solve this, we'll follow these steps:

1. **Convert the base-6 number to base 10 (decimal):**

   \[
   (523)_6 = 5 \times 6^2 + 2 \times 6^1 + 3 \times 6^0
   \]

   Calculate each term:
   \[
   5 \times 6^2 = 5 \times 36 = 180
   \]
   \[
   2 \times 6^1 = 2 \times 6 = 12
   \]
   \[
   3 \times 6^0 = 3 \times 1 = 3
   \]

   Now, add them together:
   \[
   180 + 12 + 3 = 195
   \]
   So, \( (523)_6 = 195 \) in decimal (base 10).

2. **Convert the decimal number 195 to base 9:**

   To convert 195 to base 9, divide the number by 9 and keep track of the remainders:

   \[
   195 \div 9 = 21 \text{ with a remainder of } 6
   \]
   \[
   21 \div 9 = 2 \text{ with a remainder of } 3
   \]
   \[
   2 \div 9 = 0 \text{ with a remainder of } 2
   \]

   Reading the remainders from bottom to top, \( 195 \) in base 10 is \( (236)_9 \).

So, the conversion of \( (523)_6 \) to base 9 gives \( (236)_9 \).

Would you like any details or have any questions?

Here are five related questions:
1. How do you convert a base-8 number to base-5?
2. What is the significance of different number bases in computer science?
3. Can you explain how to convert a number from base-16 to base-2?
4. What is the process for converting a decimal number to a base-7 number?
5. Why do we use base-10 (decimal) in everyday life?

**Tip:** When converting between bases, converting first to base 10 (decimal) is often the easiest approach, especially when the base is not a power of 2.

How do you convert the number (523)_6 to base 9?

This guide explains the process of converting the base-6 number (523)_6 into base 9 using step-by-step calculations. Key concepts include number systems, base conversion formulas, and positional notation. Ideal for students in Grades 9-11.

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