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.