Simplify `sqrt([(12.1)^(2) - (8.1)^(2)] / [(0.25)^(2) + (0.25) (19.95)])`

Options

A. 1
B. 2
C. 3
D. 4

Answer

To simplify the expression \( \sqrt{\frac{(12.1)^2 - (8.1)^2}{(0.25)^2 + (0.25)(19.95)}} \), we can follow these steps:

### Step 1: Identify the expression
We start with the expression:
\[ \sqrt{\frac{(12.1)^2 - (8.1)^2}{(0.25)^2 + (0.25)(19.95)}} \]

### Step 2: Apply the difference of squares formula
Recall the difference of squares formula:
\[ A^2 - B^2 = (A + B)(A - B) \]
Here, let \( A = 12.1 \) and \( B = 8.1 \). Thus, we can rewrite the numerator:
\[ (12.1)^2 - (8.1)^2 = (12.1 + 8.1)(12.1 - 8.1) \]

### Step 3: Calculate \( 12.1 + 8.1 \) and \( 12.1 - 8.1 \)
Calculating these values:
\[ 12.1 + 8.1 = 20.2 \]
\[ 12.1 - 8.1 = 4 \]
So, the numerator becomes:
\[ 20.2 \times 4 \]

### Step 4: Simplify the denominator
Now, simplify the denominator:
\[ (0.25)^2 + (0.25)(19.95) = 0.0625 + 4.9875 = 5.05 \]

### Step 5: Substitute back into the expression
Now we can substitute back into our expression:
\[ \sqrt{\frac{20.2 \times 4}{5.05}} \]

### Step 6: Simplify the fraction
Calculate the fraction:
\[ \frac{20.2 \times 4}{5.05} = \frac{80.8}{5.05} \]

### Step 7: Perform the division
Now, performing the division:
\[ 80.8 \div 5.05 = 16 \]

### Step 8: Take the square root
Finally, take the square root:
\[ \sqrt{16} = 4 \]

### Final Answer
Thus, the simplified form of the expression is:
\[ \boxed{4} \]