I keep confusing myself. Please check and show me where I am mixed up.

(8+ �ã48)/4 = (8+ �ã(2 .24))/4 =1+2�ã6
where am I missing it?

Let's break down the steps to see where the confusion lies:

Step 1: (8+ ²√48)/4
In this step, you correctly started with the expression (8+√48)/4.

Step 2: (8+√(2 × 24))/4
Here, you attempted to simplify the square root of 48. However, you made a mistake by thinking that the square root of 48 is equal to the square root of 2 times the square root of 24. This is not accurate.

Step 3: 1+2√6
Based on the incorrect simplification in Step 2, you ended up with the expression 1+2√6.

Where you went wrong:
The mistake occurred in step 2, where you incorrectly simplified the square root of 48. Instead of separating the square root of 48 into the square root of 2 times the square root of 24, you should simplify the square root of 48 as the square root of 16 times the square root of 3. This is because 48 can be factored as 16 × 3.

Corrected solution:
(8+ √48)/4 = (8+ √(16 × 3))/4 [Simplified the square root of 48]
= (8+ √16 × √3)/4
= (8+ 4√3)/4 [Simplified the square root of 16 as 4]
= 2 + √3 [Simplified (8+4√3)/4 as 2 + (√3)/1, simplified fractions]

Therefore, the correct simplified expression is 2+ √3.

Feel free to ask if you have any more questions or need further clarification!