Simplify `sqrt(25.4016)`

Options

A. 401/607
B. 104/706
C. 41/76
D. None of these

Answer

To simplify \( \sqrt{25.4016} \), we can follow these steps:

### Step 1: Remove the Decimal
To simplify the square root, we can eliminate the decimal by multiplying the number by 10,000 (since there are four digits after the decimal). This gives us:
\[ \sqrt{25.4016} = \sqrt{25.4016 \times 10,000} = \sqrt{254016} \]

### Step 2: Factor the Number
Next, we will factor \( 254016 \) to find its square root. We can start by breaking it down into smaller factors.

### Step 3: Pair the Digits
When finding the square root, we can group the digits in pairs from right to left:
- The pairs are: \( 25 \) and \( 40 \) and \( 16 \).

### Step 4: Calculate the Square Roots of Each Pair
Now we calculate the square root of each pair:
- \( \sqrt{25} = 5 \)
- \( \sqrt{40} \) is not a perfect square, but we can approximate it.
- \( \sqrt{16} = 4 \)

### Step 5: Combine the Results
Since \( \sqrt{40} \) is not a perfect square, we can express it as \( \sqrt{4 \times 10} = 2\sqrt{10} \). Thus, we can combine our results:
\[ \sqrt{254016} = 5 \times 2\sqrt{10} \times 4 = 20\sqrt{10} \]

### Step 6: Adjust for the Decimal
Since we multiplied by \( 10,000 \) earlier, we need to adjust our final answer back by taking the square root of \( 10,000 \), which is \( 100 \):
\[ \sqrt{25.4016} = \frac{20\sqrt{10}}{100} = \frac{1}{5}\sqrt{10} \]

### Final Answer
Thus, the simplified form of \( \sqrt{25.4016} \) is:
\[ \frac{1}{5}\sqrt{10} \]