Explanation:
Okay, so I need to evaluate this expression: the ceiling of (3.6) squared minus the ceiling of 3.6 squared. Let me write that down to make sure I have it right. It's ⌈(3.6)²⌉ - (⌈3.6⌉)². Alright, let me break this down step by step.

First, let me recall what the ceiling function does. The ceiling of a number is the smallest integer greater than or equal to that number. So, for example, the ceiling of 3.2 would be 4, because 4 is the next integer higher than 3.2. Similarly, the ceiling of 3.9 would also be 4, and the ceiling of 4 itself is 4. Got it.

So, the expression has two parts: the first part is the ceiling of (3.6) squared, and the second part is the ceiling of 3.6 squared. Then we subtract the second part from the first. Let me compute each part separately.

Starting with the first part: ⌈(3.6)²⌉. Let me calculate (3.6) squared first. 3.6 times 3.6. Hmm, 3 times 3 is 9, 3 times 0.6 is 1.8, 0.6 times 3 is another 1.8, and 0.6 times 0.6 is 0.36. Adding those up: 9 + 1.8 + 1.8 + 0.36. Let me compute that. 9 + 1.8 is 10.8, plus another 1.8 is 12.6, plus 0.36 is 12.96. So, (3.6)² equals 12.96. Now, taking the ceiling of that. Since 12.96 is between 12 and 13, the ceiling would be 13. So, the first part is 13.

Now, moving on to the second part: (⌈3.6⌉)². Let me compute the ceiling of 3.6 first. As I thought earlier, the ceiling of 3.6 is 4 because it's the smallest integer greater than or equal to 3.6. Then, squaring that result: 4 squared is 16. So, the second part is 16.

Now, subtracting the second part from the first part: 13 - 16. That would be... let me do the subtraction. 13 minus 16 is -3. Wait, so the result is -3? Let me check my steps again to make sure I didn't make a mistake.

First, calculating (3.6)^2. Let me verify that again. 3.6 times 3.6. Alternatively, I can think of 3.6 as 36/10. So, (36/10)^2 is 1296/100, which is 12.96. Yep, that's correct. Then ceiling of 12.96 is indeed 13. Then ceiling of 3.6 is 4, squared is 16. Then 13 - 16 is -3. That seems straightforward.

Wait, but let me make sure I didn't mix up the order of operations or something. The original expression is ⌈(3.6)^2⌉ - (⌈3.6⌉)^2. So, yes, first compute (3.6)^2, take ceiling, then compute ceiling of 3.6, square it, then subtract the second result from the first. So, 13 - 16 = -3. That seems right.

But just to be thorough, let me check if there's any alternative interpretation. For example, could someone misread the expression as ceiling of (3.6 squared minus ceiling of 3.6 squared)? But no, the parentheses make it clear. The first term is ceiling of (3.6 squared), and the second term is (ceiling of 3.6) squared. So, the way it's written, it's definitely ⌈(3.6)^2⌉ - (⌈3.6⌉)^2.

Alternatively, maybe I miscalculated the ceiling of 3.6? Let me check that again. 3.6 is between 3 and 4. The ceiling function rounds up to the next integer, so yes, ceiling of 3.6 is 4. So, squaring that gives 16. Then ceiling of 12.96 is 13. So, 13 - 16 is indeed -3.

Wait, but maybe I made a mistake in calculating (3.6)^2? Let me do that again. 3.6 times 3.6. Let me compute 3 * 3.6 first. 3 * 3.6 is 10.8. Then 0.6 * 3.6 is... 0.6 * 3 is 1.8, and 0.6 * 0.6 is 0.36, so total is 1.8 + 0.36 = 2.16. Then adding 10.8 + 2.16 gives 12.96. Yep, same result. So that's correct.

Alternatively, maybe I can use decimal multiplication. 3.6 * 3.6. Multiply 36 * 36 first, which is 1296, and since each 3.6 has one decimal place, total two decimal places, so 12.96. Yep, same thing. So that's correct.

So, ceiling of 12.96 is 13. Then ceiling of 3.6 is 4, squared is 16. 13 - 16 is -3. So, the answer is -3. Is there any chance that the ceiling function could be different? For example, if the number was exactly an integer, but 3.6 isn't. So, ceiling of 3.6 is definitely 4. And ceiling of 12.96 is 13. So, I think that's right.

Alternatively, maybe the problem is testing something else? Let me think. For example, sometimes with ceiling functions, people might confuse with floor functions, but no, the problem specifically says ceiling. So, I think my reasoning is correct.

Alternatively, maybe I need to check if there's a different order of operations? Let me check the original problem again: Evaluate ⌈(3.6)^2⌉ - (⌈3.6⌉)^2. So, the first term is ceiling of (3.6 squared), the second term is (ceiling of 3.6) squared. So, the operations are as I did. So, 13 - 16 = -3.

Alternatively, maybe the problem is written differently? For example, if it was ⌈(3.6)^2 - (⌈3.6⌉)^2⌉, but no, the original problem is written as ⌈(3.6)^2⌉ - (⌈3.6⌉)^2. So, parentheses are around the 3.6 squared in the first term, and the second term is ceiling of 3.6 squared. So, no, the way it's written, it's definitely the difference between those two terms.

Therefore, I think my answer is correct. The result is -3. Let me just write that in the box as requested.

**Final Answer**
\boxed{-3}

To evaluate the expression:

$$
\lceil (3.6)^2 \rceil - ( \lceil 3.6 \rceil )^2
$$

we proceed step by step, carefully applying the ceiling function and arithmetic operations.

---

### Step 1: Compute $ (3.6)^2 $

We calculate the square of 3.6:

$$
(3.6)^2 = 3.6 \times 3.6 = 12.96
$$

---

### Step 2: Apply the ceiling function to $ (3.6)^2 $

The ceiling function $ \lceil x \rceil $ gives the smallest integer greater than or equal to $ x $. Since $ 12.96 $ is not an integer, we round it up:

$$
\lceil 12.96 \rceil = 13
$$

---

### Step 3: Compute $ \lceil 3.6 \rceil $

Again, applying the ceiling function to 3.6:

$$
\lceil 3.6 \rceil = 4
$$

---

### Step 4: Square the result from Step 3

$$
( \lceil 3.6 \rceil )^2 = 4^2 = 16
$$

---

### Step 5: Subtract the results

Now, subtract the second result from the first:

$$
\lceil (3.6)^2 \rceil - ( \lceil 3.6 \rceil )^2 = 13 - 16 = -3
$$

---

### Final Answer

$$
\boxed{-3}
$$

[Assistant]
Explanation:
Okay, so I need to evaluate this expression: the ceiling of (3.6) squared minus the ceiling of 3.6 squared. Let me write that down to make sure I have it right. It's ⌈(3.6)²⌉ - (⌈3.6⌉)². Alright, let me break this down step by step.

First, let me recall what the ceiling function does. The ceiling of a number is the smallest integer greater than or equal to that number. So, for example, the ceiling of 3.2 would be 4, because 4 is the next integer higher than 3.2. Similarly, the ceiling of 3.9 would also be 4, and the ceiling of 4 itself is 4. Got it.

So, the expression has two parts: the first part is the ceiling of (3.6) squared, and the second part is the ceiling of 3.6 squared. Then we subtract the second part from the first. Let me compute each part separately.

Starting with the first part: ⌈(3.6)²⌉. Let me calculate (3.6) squared first. 3.6 times 3.6. Hmm, 3 times 3 is 9, 3 times 0.6 is 1.8, 0.6 times 3 is another 1.8, and 0.6 times 0.6 is 0.36. Adding those up: 9 + 1.8 + 1.8 + 0.36. Let me compute that. 9 + 1.8 is 10.8, plus another 1.8 is 12.6, plus 0.36 is 12.96. So, (3.6)² equals 12.96. Now, taking the ceiling of that. Since 12.96 is between 12 and 13, the ceiling would be 13. So, the first part is 13.

Now, moving on to the second part: (⌈3.6⌉)². Let me compute the ceiling of 3.6 first. As I thought earlier, the ceiling of 3.6 is 4 because it's the smallest integer greater than or equal to 3.6. Then, squaring that result: 4 squared is 16. So, the second part is 16.

Now, subtracting the second part from the first part: 13 - 16. That would be... let me do the subtraction. 13 minus 16 is -3. Wait, so the result is -3? Let me check my steps again to make sure I didn't make a mistake.

First, calculating (3.6)^2. Let me verify that again. 3.6 times 3.6. Alternatively, I can think of 3.6 as 36/10. So, (36/10)^2 is 1296/100, which is 12.96. Yep, that's correct. Then ceiling of 12.96 is indeed 13. Then ceiling of 3.6 is 4, squared is 16. Then 13 - 16 is -3. That seems straightforward.

Wait, but let me make sure I didn't mix up the order of operations or something. The original expression is ⌈(3.6)^2⌉ - (⌈3.6⌉)^2. So, yes, first compute (3.6)^2, take ceiling, then compute ceiling of 3.6, square it, then subtract the second result from the first. So, 13 - 16 = -3. That seems right.

But just to be thorough, let me check if there's any alternative interpretation. For example, could someone misread the expression as ceiling of (3.6 squared minus ceiling of 3.6 squared)? But no, the parentheses make it clear. The first term is ceiling of (3.6 squared), and the second term is (ceiling of 3.6) squared. So, the way it's written, it's definitely ⌈(3.6)^2⌉ - (⌈3.6⌉)^2.

Alternatively, maybe I miscalculated the ceiling of 3.6? Let me check that again. 3.6 is between 3 and 4. The ceiling function rounds up to the next integer, so yes, ceiling of 3.6 is 4. So, squaring that gives 16. Then ceiling of 12.96 is 13. So, 13 - 16 is indeed -3.

Wait, but maybe I made a mistake in calculating (3.6)^2? Let me do that again. 3.6 times 3.6. Let me compute 3 * 3.6 first. 3 * 3.6 is 10.8. Then 0.6 * 3.6 is... 0.6 * 3 is 1.8, and 0.6 * 0.6 is 0.36, so total is 1.8 + 0.36 = 2.16. Then adding 10.8 + 2.16 gives 12.96. Yep, same result. So that's correct.

Alternatively, maybe I can use decimal multiplication. 3.6 * 3.6. Multiply 36 * 36 first, which is 1296, and since each 3.6 has one decimal place, total two decimal places, so 12.96. Yep, same thing. So that's correct.

So, ceiling of 12.96 is 13. Then ceiling of 3.6 is 4, squared is 16. 13 - 16 is -3. So, the answer is -3. Is there any chance that the ceiling function could be different? For example, if the number was exactly an integer, but 3.6 isn't. So, ceiling of 3.6 is definitely 4. And ceiling of 12.96 is 13. So, I think that's right.

Alternatively, maybe the problem is testing something else? Let me think. For example, sometimes with ceiling functions, people might confuse with floor functions, but no, the problem specifically says ceiling. So, I think my reasoning is correct.

Alternatively, maybe I need to check if there's a different order of operations? Let me check the original problem again: Evaluate ⌈(3.6)^2⌉ - (⌈3.6⌉)^2. So, the first term is ceiling of (3.6 squared), the second term is (ceiling of 3.6) squared. So, the operations are as I did. So, 13 - 16 = -3.

Alternatively, maybe the problem is written differently? For example, if it was ⌈(3.6)^2 - (⌈3.6⌉)^2⌉, but no, the original problem is written as ⌈(3.6)^2⌉ - (⌈3.6⌉)^2. So, parentheses are around the 3.6 squared in the first term, and the second term is ceiling of 3.6 squared. So, no, the way it's written, it's definitely the difference between those two terms.

Therefore, I think my answer is correct. The result is -3. Let me just write that in the box as requested.

**Final Answer**
\boxed{-3}

To evaluate the expression:

$$
\lceil (3.6)^2 \rceil - ( \lceil 3.6 \rceil )^2
$$

we proceed step by step, carefully applying the ceiling function and arithmetic operations.

---

### Step 1: Compute $ (3.6)^2 $

We calculate the square of 3.6:

$$
(3.6)^2 = 3.6 \times 3.6 = 12.96
$$

---

### Step 2: Apply the ceiling function to $ (3.6)^2 $

The ceiling function $ \lceil x \rceil $ gives the smallest integer greater than or equal to $ x $. Since $ 12.96 $ is not an integer, we round it up:

$$
\lceil 12.96 \rceil = 13
$$

---

### Step 3: Compute $ \lceil 3.6 \rceil $

Again, applying the ceiling function to 3.6:

$$
\lceil 3.6 \rceil = 4
$$

---

### Step 4: Square the result from Step 3

$$
( \lceil 3.6 \rceil )^2 = 4^2 = 16
$$

---

### Step 5: Subtract the results

Now, subtract the second result from the first:

$$
\lceil (3.6)^2 \rceil - ( \lceil 3.6 \rceil )^2 = 13 - 16 = -3
$$

---

### Final Answer

$$
\boxed{-3}
$$

Solution: $\lceil (3.6)^2 \rceil = \lceil 12.96 \rceil = 13$ because the least integer greater than $12.96$ is $13$. $( \lceil 3.6 \rceil ) ^2 = 4^2 = 16$ because the least integer greater than $3.6$ is $4$. Therefore, the answer is $13-16=\boxed{-3}$.