Find ?
`sqrt(5329)`

Answer

To find the square root of 5329, we can use the long division method. Here’s a step-by-step solution:

### Step 1: Pair the digits
First, we need to pair the digits of the number starting from the right. For 5329, we can pair it as (53)(29).

### Step 2: Find the largest square less than the first pair
Now, we look at the first pair, which is 53. We need to find the largest integer whose square is less than or equal to 53.

- \(7^2 = 49\) (which is less than 53)
- \(8^2 = 64\) (which is greater than 53)

So, we take 7.

### Step 3: Subtract and bring down the next pair
Now, we subtract \(49\) from \(53\):
\[ 53 - 49 = 4 \]
Next, we bring down the next pair (29), making it 429.

### Step 4: Double the quotient and find the next digit
Next, we double the quotient we found (which is 7):
\[ 2 \times 7 = 14 \]
Now, we need to find a digit (let's call it \(x\)) such that:
\[ (140 + x) \times x \leq 429 \]
We can test values for \(x\):
- If \(x = 3\):
\[ (140 + 3) \times 3 = 143 \times 3 = 429 \]
This works perfectly since it equals 429.

### Step 5: Finalize the answer
Now we subtract:
\[ 429 - 429 = 0 \]
Since we have reached 0, we can conclude that the square root of 5329 is:
\[ \sqrt{5329} = 73 \]

### Final Answer
The square root of 5329 is \(73\).

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