To solve the given mathematical expression $\sqrt{12.0409}-\sqrt{1.2544}$, we need to calculate the square root of each number and then find their difference.

Calculating the Square Root of 12.0409

We are looking for a number which, when multiplied by itself, equals 12.0409. Let's estimate. We know that $3^2 = 9$ and $4^2 = 16$. Since 12.0409 is between 9 and 16, its square root will be between 3 and 4. The number 12.0409 ends with the digit 9, which means its square root must end with either 3 or 7.

Let's try 3.4:

$$3.4 \times 3.4 = 11.56$$

Let's try 3.5:

$$3.5 \times 3.5 = 12.25$$

Since 12.0409 is closer to 12.25, let's try a number slightly less than 3.5, ending in 7. Let's try 3.47:

$$3.47 \times 3.47 = 12.0409$$

So, the square root of 12.0409 is 3.47.

Calculating the Square Root of 1.2544

Similarly, we need to find the square root of 1.2544. We know that $1^2 = 1$ and $2^2 = 4$. Since 1.2544 is between 1 and 4, its square root will be between 1 and 2. The number 1.2544 ends with the digit 4, which means its square root must end with either 2 or 8.

Let's try 1.1:

$$1.1 \times 1.1 = 1.21$$

Let's try 1.2:

$$1.2 \times 1.2 = 1.44$$

Since 1.2544 is between 1.21 and 1.44, let's try a number ending in 2, like 1.12:

$$1.12 \times 1.12 = 1.2544$$

So, the square root of 1.2544 is 1.12.

Finding the Difference

Now, we subtract the second square root from the first:

$$ \sqrt{12.0409} - \sqrt{1.2544} = 3.47 - 1.12 $$

$$ 3.47 - 1.12 = 2.35 $$

Therefore, the value of the expression is 2.35.