Evaluate:
`sqrt (92416)`

Answer

To evaluate \( \sqrt{92416} \), we can follow these steps:

### Step 1: Pair the digits
We start by pairing the digits of the number from right to left. For \( 92416 \), we can pair it as \( 92 | 41 | 6 \).

### Step 2: Find the largest square less than or equal to the first pair
The first pair is \( 92 \). The largest perfect square less than or equal to \( 92 \) is \( 81 \) (which is \( 9^2 \)). So, we take \( 9 \) as the first digit of our square root.

### Step 3: Subtract and bring down the next pair
Subtract \( 81 \) from \( 92 \):
\[ 92 - 81 = 11 \]
Now, bring down the next pair \( 41 \) to get \( 1141 \).

### Step 4: Double the current result
Double the current result (which is \( 9 \)):
\[ 2 \times 9 = 18 \]

### Step 5: Find the next digit
Now we need to find a digit \( x \) such that \( (180 + x) \times x \) is less than or equal to \( 1141 \).

Testing \( x = 6 \):
\[ (180 + 6) \times 6 = 186 \times 6 = 1116 \]
This is less than \( 1141 \).

Testing \( x = 7 \):
\[ (180 + 7) \times 7 = 187 \times 7 = 1309 \]
This is greater than \( 1141 \).

So, we take \( x = 6 \).

### Step 6: Subtract and bring down the next pair
Now we have:
\[ 1141 - 1116 = 25 \]
Bring down the next pair \( 00 \) (since there are no more digits, we can consider it as \( 00 \)) to get \( 2500 \).

### Step 7: Double the current result again
Now double the current result \( 96 \) (which is \( 9 \) and \( 6 \)):
\[ 2 \times 96 = 192 \]

### Step 8: Find the next digit
Now we need to find a digit \( y \) such that \( (1920 + y) \times y \) is less than or equal to \( 2500 \).

Testing \( y = 1 \):
\[ (1920 + 1) \times 1 = 1921 \]
This is less than \( 2500 \).

Testing \( y = 2 \):
\[ (1920 + 2) \times 2 = 1922 \times 2 = 3844 \]
This is greater than \( 2500 \).

So, we take \( y = 1 \).

### Step 9: Final result
Now we have:
\[ 2500 - 1921 = 579 \]
Since we have no more pairs to bring down, we conclude that:
\[ \sqrt{92416} = 304 \]

### Final Answer:
\[ \sqrt{92416} = 304 \]