To convert the number \(1265_{\text{base}\,11}\) to decimal (base 10), we expand it using the powers of 11. Here's the step-by-step process:

\[
1265_{\text{base}\,11} = 1 \times 11^3 + 2 \times 11^2 + 6 \times 11^1 + 5 \times 11^0
\]

Let's compute each term:

\[
1 \times 11^3 = 1 \times 1331 = 1331
\]
\[
2 \times 11^2 = 2 \times 121 = 242
\]
\[
6 \times 11^1 = 6 \times 11 = 66
\]
\[
5 \times 11^0 = 5 \times 1 = 5
\]

Now, sum all the values:

\[
1331 + 242 + 66 + 5 = 1644
\]

Therefore, \(1265_{\text{base}\,11} = 1644_{\text{base}\,10}\).

Would you like more details or have any questions? Here are five related questions to explore:

1. How do you convert a number from decimal to base 11?
2. What is the general formula for converting any base to decimal?
3. How would you convert \(1644_{\text{base}\,10}\) back to base 11?
4. What is \(1011_{\text{base}\,2}\) in decimal?
5. How do you perform arithmetic operations (addition, subtraction) in base 11?

**Tip:** When converting from any base to decimal, remember to multiply each digit by the base raised to the power of its position from right to left, starting at 0.

Learn how to convert 1265 from base 11 to decimal using the base conversion formula. This problem demonstrates powers of numbers and expands on key number system concepts. Ideal for students in Grades 9-10.

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