Evaluate:
`sqrt (75.69)`

Answer

To evaluate \( \sqrt{75.69} \), we can follow these steps:

### Step 1: Remove the Decimal
To make calculations easier, we can remove the decimal by multiplying the number by 100. Thus, we rewrite \( 75.69 \) as \( 7569 \) (since \( 75.69 \times 100 = 7569 \)).

### Step 2: Find the Square Root
Now, we need to find the square root of \( 7569 \). We can do this through prime factorization or long division. Here, we will use the prime factorization method.

### Step 3: Prime Factorization of 7569
To find the prime factors of \( 7569 \):
- Start dividing \( 7569 \) by the smallest prime number, which is \( 3 \):
- \( 7569 \div 3 = 2523 \)
- Continue dividing \( 2523 \) by \( 3 \):
- \( 2523 \div 3 = 841 \)
- Now, \( 841 \) is not divisible by \( 3 \), so we check the next prime number, which is \( 29 \):
- \( 841 \div 29 = 29 \)
- Thus, the prime factorization of \( 7569 \) is \( 3^2 \times 29^2 \).

### Step 4: Taking the Square Root
Now we can take the square root of \( 7569 \):
- \( \sqrt{7569} = \sqrt{3^2 \times 29^2} = 3 \times 29 = 87 \).

### Step 5: Applying the Decimal
Since we multiplied by \( 100 \) in the first step, we need to apply the decimal back:
- The square root of \( 75.69 \) is \( 87 \).

### Final Answer
Thus, \( \sqrt{75.69} = 8.7 \).