Define `[[x]]` to be the largest integer less than x. what is the value of `[[sqrt(75)]]`?

Answer

To find the value of `[[sqrt(75)]]`, we will follow these steps:

### Step 1: Calculate `sqrt(75)`
First, we need to find the square root of 75. We can simplify this by expressing 75 as a product of its prime factors.

\[ 75 = 25 \times 3 = 5^2 \times 3 \]

Thus, we can write:

\[ \sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3} \]

### Step 2: Approximate `sqrt(3)`
Next, we need to approximate the value of `sqrt(3)`. The approximate value of `sqrt(3)` is about 1.732.

### Step 3: Calculate `5 * sqrt(3)`
Now we can calculate `5 * sqrt(3)`:

\[ 5\sqrt{3} \approx 5 \times 1.732 \approx 8.66 \]

### Step 4: Determine the largest integer less than `8.66`
Now we need to find the largest integer that is less than `8.66`. The integers less than `8.66` are:

\[ ..., -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8 \]

The largest integer in this list is `8`.

### Conclusion
Thus, the value of `[[sqrt(75)]]` is:

\[ \boxed{8} \]

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