Explanation:
Okay, so I need to find the square root of f(f(f(f(1)))), where f(x) is 2^x. Let me break this down step by step. First, let me make sure I understand the problem correctly. The function f is defined as f(x) = 2^x, which is an exponential function with base 2. The question is asking for the square root of the result of applying this function four times in succession, starting with the input 1. So, it's like f(f(f(f(1)))), and then take the square root of that.

Let me write that out more clearly. Let me denote each step:

First, compute f(1). Then, take that result and plug it into f again to get f(f(1)). Then take that result and plug it into f again to get f(f(f(1))), and then one more time to get f(f(f(f(1)))). After that, take the square root of that final value.

So, step by step:

1. Compute f(1) = 2^1 = 2.
2. Then compute f(f(1)) = f(2) = 2^2 = 4.
3. Next, compute f(f(f(1))) = f(4) = 2^4 = 16.
4. Then compute f(f(f(f(1)))) = f(16) = 2^16.
5. Finally, take the square root of 2^16.

Wait, let me check that again. Let me make sure I didn't skip any steps or make a miscalculation. Let me go through each step again.

Starting with f(1):

f(1) = 2^1 = 2. That seems straightforward.

Then f(f(1)) is f(2). Since f(2) = 2^2 = 4. Correct.

Then f(f(f(1))) is f(4). So f(4) = 2^4. Let me compute 2^4. 2*2=4, 4*2=8, 8*2=16. So 2^4 is 16. Correct.

Then f(f(f(f(1)))) is f(16). So f(16) = 2^16. Now, 2^16 is a larger number. Let me recall that 2^10 is 1024, so 2^16 is 2^10 * 2^6 = 1024 * 64. Let me compute that. 1024 * 64. Well, 1000*64 = 64,000 and 24*64 = 1,536. So adding those together: 64,000 + 1,536 = 65,536. Therefore, 2^16 is 65,536.

So then, the problem asks for the square root of that. So sqrt(65,536). Let me compute that.

But maybe there's a smarter way than just calculating 65,536 and then taking the square root. Let me think. Since 2^16 is (2^8)^2, because (2^8)^2 = 2^(8*2) = 2^16. Therefore, sqrt(2^16) is 2^(16/2) = 2^8. Which is 256.

Wait, that seems much simpler. So instead of calculating 2^16 first, which is 65,536, and then taking the square root, I can recognize that sqrt(2^16) is 2^8. Therefore, the answer is 2^8, which is 256.

But let me verify that. Let me check if sqrt(2^16) is indeed 2^8. Since sqrt(a^b) is a^(b/2), so sqrt(2^16) = 2^(16/2) = 2^8. Yes, that's correct. Therefore, the square root of 2^16 is 2^8, which is 256.

Therefore, the answer is 256.

But let me just make sure I didn't make a mistake in the earlier steps. Let me retrace:

Start with f(1) = 2^1 = 2. Then f(f(1)) = f(2) = 2^2 = 4. Then f(f(f(1))) = f(4) = 2^4 = 16. Then f(f(f(f(1)))) = f(16) = 2^16. Then sqrt(2^16) = 2^8 = 256.

Yes, that seems correct.

Alternatively, maybe I can think of the entire expression as follows. Let me denote each application of f:

Let me define:

Let’s let’s denote:

First application: f(1) = 2^1 = 2.

Second application: f(f(1)) = f(2) = 2^2 = 4.

Third application: f(f(f(1))) = f(4) = 2^4 = 16.

Fourth application: f(f(f(f(1)))) = f(16) = 2^16.

Therefore, sqrt(f(f(f(f(1))))) = sqrt(2^16) = 2^(16/2) = 2^8 = 256.

Yes, that seems consistent.

Alternatively, maybe I can think of the exponents. Since each time we apply f, we are exponentiating the previous result with base 2. So starting with 1, then 2^1, then 2^(2^1), then 2^(2^(2^1)), etc. Wait, but hold on, is that the case?

Wait, no. Wait, f(x) = 2^x. So f(f(x)) = 2^(f(x)) = 2^(2^x). Similarly, f(f(f(x))) = 2^(f(f(x))) = 2^(2^(2^x)), and so on. Therefore, applying f four times would be 2^(2^(2^(2^x))). But in this case, starting with x=1. So f(f(f(f(1)))) would be 2^(2^(2^(2^1))). Let me compute that.

Let me compute the innermost exponent first. Starting with 1:

First, 2^1 = 2.

Then, the next layer is 2^2 = 4.

Then, the next layer is 2^4 = 16.

Then, the next layer is 2^16 = 65536.

Therefore, f(f(f(f(1)))) = 2^(2^(2^(2^1))) = 2^16 = 65536. Therefore, sqrt(65536) is 256.

So that's another way to think about it, breaking down the exponents.

Therefore, regardless of the approach, the answer is 256.

But just to make sure, let me compute sqrt(65536) directly. Let me recall that 256 squared is 65536. Let me check:

256 * 256. Let me compute 256 * 256.

Well, 256 is 2^8, so 2^8 * 2^8 = 2^16 = 65536. Therefore, sqrt(65536) is indeed 256.

Alternatively, if I didn't remember that 256 squared is 65536, I could compute it step by step. Let me try:

256 * 256:

First, break it down as (200 + 56) * (200 + 56). But that might be complicated. Alternatively, note that 256 * 256 = (250 + 6) * (250 + 6) = 250^2 + 2*250*6 + 6^2 = 62500 + 3000 + 36 = 62500 + 3000 is 65500, plus 36 is 65536. Yes, that works.

Alternatively, 256 * 256. Since 256 * 2 = 512, 256 * 4 = 1024, 256 * 8 = 2048, 256 * 16 = 4096, 256 * 32 = 8192, 256 * 64 = 16384, 256 * 128 = 32768, 256 * 256 = 65536.

Alternatively, since 256 is 2^8, so 2^8 * 2^8 = 2^16, which is 65536.

Therefore, sqrt(65536) is 2^8 = 256.

Therefore, regardless of the method, the answer is 256.

So, putting it all together, the steps are:

Start with 1.

Apply f once: 2^1 = 2.

Apply f again: 2^2 = 4.

Apply f again: 2^4 = 16.

Apply f again: 2^16 = 65536.

Take the square root: sqrt(65536) = 256.

Therefore, the answer is 256.

I think that's solid. Let me check once more if I miscounted the number of function applications. The problem says sqrt(f(f(f(f(1))))). So that's four f's. Let me count:

First f: f(1)

Second f: f(f(1))

Third f: f(f(f(1)))

Fourth f: f(f(f(f(1))))

Yes, four applications. So that's correct.

Alternatively, if I had miscounted and thought it was three applications, then the answer would be different, but no, the problem says four.

Another way to think about it: if I had to write f(f(f(f(1)))), that's f^4(1), where f^4 denotes the fourth iterate of f.

So, in terms of exponents, each iteration is exponentiating the previous result with base 2. So starting with 1:

f^1(1) = 2^1 = 2

f^2(1) = 2^2 = 4

f^3(1) = 2^4 = 16

f^4(1) = 2^16 = 65536

Then sqrt(65536) = 256.

Yes, that's consistent.

Alternatively, if I wanted to write this in terms of exponents, maybe I can find a pattern. Let me see:

Let me denote f^n(1) as the nth iterate. Then:

f^1(1) = 2^1 = 2

f^2(1) = 2^2 = 4

f^3(1) = 2^4 = 16

f^4(1) = 2^16 = 65536

So, each time, the exponent is the previous result. So, the exponents are 1, 2, 4, 16, ... which is 2^0, 2^1, 2^2, 2^4, ... Wait, no:

Wait, f^1(1) = 2^1 = 2^1

f^2(1) = 2^2 = 2^(2^1)

f^3(1) = 2^(2^2) = 2^(2^2)

f^4(1) = 2^(2^4) = 2^(2^(2^2))

Wait, so in general, f^n(1) = 2^(f^(n-1)(1))? Wait, no. Wait, f^1(1) = 2^1. Then f^2(1) = f(f(1)) = f(2) = 2^2. Then f^3(1) = f(f(f(1))) = f(4) = 2^4. Then f^4(1) = f(16) = 2^16. So, in general, f^n(1) = 2^(f^(n-1)(1)). Therefore, each time, the exponent is the previous value.

So, for n=1: 2^1

n=2: 2^(2^1)

n=3: 2^(2^(2^1))

n=4: 2^(2^(2^(2^1)))

Which is a tower of exponents. However, in this case, since we are only going up to n=4, it's manageable.

But regardless, the value of f^4(1) is 2^16, so sqrt(2^16) is 2^8 = 256.

Therefore, I think I've verified this multiple ways.

Another angle: maybe using logarithms? Although that might be overcomplicating, but let me try.

Suppose I want to compute sqrt(f(f(f(f(1))))). Let me denote that as sqrt(2^16). Taking the square root is the same as raising to the power of 1/2. Therefore, (2^16)^(1/2) = 2^(16*(1/2)) = 2^8 = 256.

Yes, that's straightforward.

Alternatively, if I didn't recognize that 2^16 is a perfect square, I could factor it. 2^16 is (2^8)^2, so sqrt(2^16) is 2^8.

Therefore, regardless of the method, the answer is 256.

I think that's thorough. I don't see any mistakes in my reasoning. Each step checks out, and multiple methods lead to the same conclusion. Therefore, I can be confident that the answer is 256.

**Final Answer**
\boxed{256}

To solve the expression
$$
\sqrt{f(f(f(f(1))))},
$$
where $ f(x) = 2^x $, we proceed step by step, applying the function repeatedly and simplifying at each stage.

---

### Step 1: Compute $ f(1) $

$$
f(1) = 2^1 = 2
$$

---

### Step 2: Compute $ f(f(1)) = f(2) $

$$
f(2) = 2^2 = 4
$$

---

### Step 3: Compute $ f(f(f(1))) = f(4) $

$$
f(4) = 2^4 = 16
$$

---

### Step 4: Compute $ f(f(f(f(1)))) = f(16) $

$$
f(16) = 2^{16}
$$

---

### Step 5: Take the square root of $ 2^{16} $

$$
\sqrt{2^{16}} = (2^{16})^{1/2} = 2^{16/2} = 2^8
$$

Now compute $ 2^8 $:

$$
2^8 = 256
$$

---

### Final Answer

$$
\boxed{256}
$$

[Assistant]
Explanation:
Okay, so I need to find the square root of f(f(f(f(1)))), where f(x) is 2^x. Let me break this down step by step. First, let me make sure I understand the problem correctly. The function f is defined as f(x) = 2^x, which is an exponential function with base 2. The question is asking for the square root of the result of applying this function four times in succession, starting with the input 1. So, it's like f(f(f(f(1)))), and then take the square root of that.

Let me write that out more clearly. Let me denote each step:

First, compute f(1). Then, take that result and plug it into f again to get f(f(1)). Then take that result and plug it into f again to get f(f(f(1))), and then one more time to get f(f(f(f(1)))). After that, take the square root of that final value.

So, step by step:

1. Compute f(1) = 2^1 = 2.
2. Then compute f(f(1)) = f(2) = 2^2 = 4.
3. Next, compute f(f(f(1))) = f(4) = 2^4 = 16.
4. Then compute f(f(f(f(1)))) = f(16) = 2^16.
5. Finally, take the square root of 2^16.

Wait, let me check that again. Let me make sure I didn't skip any steps or make a miscalculation. Let me go through each step again.

Starting with f(1):

f(1) = 2^1 = 2. That seems straightforward.

Then f(f(1)) is f(2). Since f(2) = 2^2 = 4. Correct.

Then f(f(f(1))) is f(4). So f(4) = 2^4. Let me compute 2^4. 2*2=4, 4*2=8, 8*2=16. So 2^4 is 16. Correct.

Then f(f(f(f(1)))) is f(16). So f(16) = 2^16. Now, 2^16 is a larger number. Let me recall that 2^10 is 1024, so 2^16 is 2^10 * 2^6 = 1024 * 64. Let me compute that. 1024 * 64. Well, 1000*64 = 64,000 and 24*64 = 1,536. So adding those together: 64,000 + 1,536 = 65,536. Therefore, 2^16 is 65,536.

So then, the problem asks for the square root of that. So sqrt(65,536). Let me compute that.

But maybe there's a smarter way than just calculating 65,536 and then taking the square root. Let me think. Since 2^16 is (2^8)^2, because (2^8)^2 = 2^(8*2) = 2^16. Therefore, sqrt(2^16) is 2^(16/2) = 2^8. Which is 256.

Wait, that seems much simpler. So instead of calculating 2^16 first, which is 65,536, and then taking the square root, I can recognize that sqrt(2^16) is 2^8. Therefore, the answer is 2^8, which is 256.

But let me verify that. Let me check if sqrt(2^16) is indeed 2^8. Since sqrt(a^b) is a^(b/2), so sqrt(2^16) = 2^(16/2) = 2^8. Yes, that's correct. Therefore, the square root of 2^16 is 2^8, which is 256.

Therefore, the answer is 256.

But let me just make sure I didn't make a mistake in the earlier steps. Let me retrace:

Start with f(1) = 2^1 = 2. Then f(f(1)) = f(2) = 2^2 = 4. Then f(f(f(1))) = f(4) = 2^4 = 16. Then f(f(f(f(1)))) = f(16) = 2^16. Then sqrt(2^16) = 2^8 = 256.

Yes, that seems correct.

Alternatively, maybe I can think of the entire expression as follows. Let me denote each application of f:

Let me define:

Let’s let’s denote:

First application: f(1) = 2^1 = 2.

Second application: f(f(1)) = f(2) = 2^2 = 4.

Third application: f(f(f(1))) = f(4) = 2^4 = 16.

Fourth application: f(f(f(f(1)))) = f(16) = 2^16.

Therefore, sqrt(f(f(f(f(1))))) = sqrt(2^16) = 2^(16/2) = 2^8 = 256.

Yes, that seems consistent.

Alternatively, maybe I can think of the exponents. Since each time we apply f, we are exponentiating the previous result with base 2. So starting with 1, then 2^1, then 2^(2^1), then 2^(2^(2^1)), etc. Wait, but hold on, is that the case?

Wait, no. Wait, f(x) = 2^x. So f(f(x)) = 2^(f(x)) = 2^(2^x). Similarly, f(f(f(x))) = 2^(f(f(x))) = 2^(2^(2^x)), and so on. Therefore, applying f four times would be 2^(2^(2^(2^x))). But in this case, starting with x=1. So f(f(f(f(1)))) would be 2^(2^(2^(2^1))). Let me compute that.

Let me compute the innermost exponent first. Starting with 1:

First, 2^1 = 2.

Then, the next layer is 2^2 = 4.

Then, the next layer is 2^4 = 16.

Then, the next layer is 2^16 = 65536.

Therefore, f(f(f(f(1)))) = 2^(2^(2^(2^1))) = 2^16 = 65536. Therefore, sqrt(65536) is 256.

So that's another way to think about it, breaking down the exponents.

Therefore, regardless of the approach, the answer is 256.

But just to make sure, let me compute sqrt(65536) directly. Let me recall that 256 squared is 65536. Let me check:

256 * 256. Let me compute 256 * 256.

Well, 256 is 2^8, so 2^8 * 2^8 = 2^16 = 65536. Therefore, sqrt(65536) is indeed 256.

Alternatively, if I didn't remember that 256 squared is 65536, I could compute it step by step. Let me try:

256 * 256:

First, break it down as (200 + 56) * (200 + 56). But that might be complicated. Alternatively, note that 256 * 256 = (250 + 6) * (250 + 6) = 250^2 + 2*250*6 + 6^2 = 62500 + 3000 + 36 = 62500 + 3000 is 65500, plus 36 is 65536. Yes, that works.

Alternatively, 256 * 256. Since 256 * 2 = 512, 256 * 4 = 1024, 256 * 8 = 2048, 256 * 16 = 4096, 256 * 32 = 8192, 256 * 64 = 16384, 256 * 128 = 32768, 256 * 256 = 65536.

Alternatively, since 256 is 2^8, so 2^8 * 2^8 = 2^16, which is 65536.

Therefore, sqrt(65536) is 2^8 = 256.

Therefore, regardless of the method, the answer is 256.

So, putting it all together, the steps are:

Start with 1.

Apply f once: 2^1 = 2.

Apply f again: 2^2 = 4.

Apply f again: 2^4 = 16.

Apply f again: 2^16 = 65536.

Take the square root: sqrt(65536) = 256.

Therefore, the answer is 256.

I think that's solid. Let me check once more if I miscounted the number of function applications. The problem says sqrt(f(f(f(f(1))))). So that's four f's. Let me count:

First f: f(1)

Second f: f(f(1))

Third f: f(f(f(1)))

Fourth f: f(f(f(f(1))))

Yes, four applications. So that's correct.

Alternatively, if I had miscounted and thought it was three applications, then the answer would be different, but no, the problem says four.

Another way to think about it: if I had to write f(f(f(f(1)))), that's f^4(1), where f^4 denotes the fourth iterate of f.

So, in terms of exponents, each iteration is exponentiating the previous result with base 2. So starting with 1:

f^1(1) = 2^1 = 2

f^2(1) = 2^2 = 4

f^3(1) = 2^4 = 16

f^4(1) = 2^16 = 65536

Then sqrt(65536) = 256.

Yes, that's consistent.

Alternatively, if I wanted to write this in terms of exponents, maybe I can find a pattern. Let me see:

Let me denote f^n(1) as the nth iterate. Then:

f^1(1) = 2^1 = 2

f^2(1) = 2^2 = 4

f^3(1) = 2^4 = 16

f^4(1) = 2^16 = 65536

So, each time, the exponent is the previous result. So, the exponents are 1, 2, 4, 16, ... which is 2^0, 2^1, 2^2, 2^4, ... Wait, no:

Wait, f^1(1) = 2^1 = 2^1

f^2(1) = 2^2 = 2^(2^1)

f^3(1) = 2^(2^2) = 2^(2^2)

f^4(1) = 2^(2^4) = 2^(2^(2^2))

Wait, so in general, f^n(1) = 2^(f^(n-1)(1))? Wait, no. Wait, f^1(1) = 2^1. Then f^2(1) = f(f(1)) = f(2) = 2^2. Then f^3(1) = f(f(f(1))) = f(4) = 2^4. Then f^4(1) = f(16) = 2^16. So, in general, f^n(1) = 2^(f^(n-1)(1)). Therefore, each time, the exponent is the previous value.

So, for n=1: 2^1

n=2: 2^(2^1)

n=3: 2^(2^(2^1))

n=4: 2^(2^(2^(2^1)))

Which is a tower of exponents. However, in this case, since we are only going up to n=4, it's manageable.

But regardless, the value of f^4(1) is 2^16, so sqrt(2^16) is 2^8 = 256.

Therefore, I think I've verified this multiple ways.

Another angle: maybe using logarithms? Although that might be overcomplicating, but let me try.

Suppose I want to compute sqrt(f(f(f(f(1))))). Let me denote that as sqrt(2^16). Taking the square root is the same as raising to the power of 1/2. Therefore, (2^16)^(1/2) = 2^(16*(1/2)) = 2^8 = 256.

Yes, that's straightforward.

Alternatively, if I didn't recognize that 2^16 is a perfect square, I could factor it. 2^16 is (2^8)^2, so sqrt(2^16) is 2^8.

Therefore, regardless of the method, the answer is 256.

I think that's thorough. I don't see any mistakes in my reasoning. Each step checks out, and multiple methods lead to the same conclusion. Therefore, I can be confident that the answer is 256.

**Final Answer**
\boxed{256}

To solve the expression
$$
\sqrt{f(f(f(f(1))))},
$$
where $ f(x) = 2^x $, we proceed step by step, applying the function repeatedly and simplifying at each stage.

---

### Step 1: Compute $ f(1) $

$$
f(1) = 2^1 = 2
$$

---

### Step 2: Compute $ f(f(1)) = f(2) $

$$
f(2) = 2^2 = 4
$$

---

### Step 3: Compute $ f(f(f(1))) = f(4) $

$$
f(4) = 2^4 = 16
$$

---

### Step 4: Compute $ f(f(f(f(1)))) = f(16) $

$$
f(16) = 2^{16}
$$

---

### Step 5: Take the square root of $ 2^{16} $

$$
\sqrt{2^{16}} = (2^{16})^{1/2} = 2^{16/2} = 2^8
$$

Now compute $ 2^8 $:

$$
2^8 = 256
$$

---

### Final Answer

$$
\boxed{256}
$$

Solution: We find that $f(1) = 2^1 = 2.$ Then, $f(f(1)) = f(2) = 2^2 = 4$ and $f(f(f(1))) = f(4) = 2^4 = 16.$ Therefore, $f(f(f(f(1)))) = f(16) = 2^{16}$ and so $\sqrt{f(f(f(f(1))))} = \sqrt{2^{16}} = 2^8 = \boxed{256}.$


Explanation:
Okay, so I need to find the integer value of x in the arithmetic sequence 3², x, 3⁴. Let me start by recalling what an arithmetic sequence is. An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. That means if I have three terms in an arithmetic sequence, the middle term is the average of the first and third terms.

First, let me write down the given terms. The first term is 3 squared, which is 9. The second term is x, which is what we need to find. The third term is 3 to the fourth power, which is 81. So the sequence is 9, x, 81.

Since it's an arithmetic sequence, the difference between the first and second term should be the same as the difference between the second and third term. Let me denote the common difference as d. So, the difference between the first and second term is x - 9, and the difference between the second and third term is 81 - x. Since these differences are equal, I can set up the equation:

x - 9 = 81 - x

Now, I need to solve for x. Let me do that step by step.

First, I can add x to both sides of the equation to get rid of the -x on the right side.

x - 9 + x = 81 - x + x

Simplifying both sides:

2x - 9 = 81

Next, I need to isolate the term with x. So, I'll add 9 to both sides:

2x - 9 + 9 = 81 + 9

Which simplifies to:

2x = 90

Now, to solve for x, I divide both sides by 2:

x = 90 / 2

x = 45

Wait, let me check if that makes sense. If the first term is 9, the second term is 45, and the third term is 81, then the common difference should be 45 - 9 = 36, and 81 - 45 = 36. Yes, that's correct. Both differences are 36, so it is an arithmetic sequence.

But let me make sure I didn't make any mistakes in my calculations. Let me go through the steps again.

Original equation: x - 9 = 81 - x

Adding x to both sides: x + x - 9 = 81

Which is 2x - 9 = 81

Adding 9 to both sides: 2x = 90

Divide by 2: x = 45. Yep, that seems right.

Alternatively, since in an arithmetic sequence the middle term is the average of the first and third terms, I can use the formula for the nth term. For three terms, the second term is the average of the first and third. So:

x = (first term + third term)/2 = (9 + 81)/2 = 90/2 = 45.

Same result. So that confirms it.

Is there another way to approach this problem? Maybe by considering the positions of the terms? Let me think. In an arithmetic sequence, each term is obtained by adding the common difference to the previous term. So, if the first term is a₁ = 9, then the second term a₂ = a₁ + d, and the third term a₃ = a₂ + d = a₁ + 2d.

Given that a₃ is 81, so:

a₁ + 2d = 81

We know a₁ is 9, so:

9 + 2d = 81

Subtract 9 from both sides:

2d = 72

Divide by 2:

d = 36

Then, the second term a₂ = a₁ + d = 9 + 36 = 45.

Same answer again. So, regardless of the method, I get x = 45.

Is there a possibility that the question is referring to something else? For example, maybe the sequence is not arithmetic but geometric? Wait, the question specifically says arithmetic sequence, so no. But just to check, if it were geometric, the ratio between terms would be constant. Let me see what would happen in that case.

In a geometric sequence, the middle term squared would be equal to the product of the first and third terms. So, x² = 9 * 81 = 729. Then x would be sqrt(729) = 27 or -27. But since the question says arithmetic sequence, not geometric, so that's not relevant here.

But just to make sure, the problem says arithmetic sequence, so we can ignore the geometric possibility.

Another check: Let me verify the arithmetic sequence with x = 45. The terms are 9, 45, 81. The difference between 9 and 45 is 36, and between 45 and 81 is also 36. So yes, that's an arithmetic sequence with common difference 36.

Therefore, I think x = 45 is the correct answer.

Wait, but the problem says "the integer value of x". Well, 45 is an integer, so that's fine.

Is there any other integer that could satisfy this? Let me think. Suppose someone thought that maybe the sequence is in some other form? For example, maybe the exponents are in arithmetic sequence? But the problem says the sequence is 3², x, 3⁴. So the terms themselves are 3², x, 3⁴. So the terms are 9, x, 81. So unless there's some other interpretation, but I don't think so.

Alternatively, maybe the exponents are in arithmetic sequence? Let me check. If the exponents were in arithmetic sequence, then 2, x, 4. Then the middle term would be (2 + 4)/2 = 3. So x would be 3, and then the terms would be 3², 3³, 3⁴. But that would be a geometric sequence with common ratio 3, not arithmetic. However, the problem says the sequence is arithmetic, so that's not the case.

Therefore, the original interpretation is correct. The terms themselves are in arithmetic sequence, so x is 45.

Another way to think about it: The difference between the first and third term is 81 - 9 = 72. In an arithmetic sequence, the difference between the first and third term is twice the common difference. So 2d = 72, so d = 36. Then, the second term is first term + d = 9 + 36 = 45.

All methods lead to the same answer. Therefore, I can be confident that x is 45.

Just to recap:

Given arithmetic sequence: 3², x, 3⁴

Convert to numbers: 9, x, 81

Common difference d = x - 9 = 81 - x

Solving gives x = 45

Check: 9, 45, 81 with differences 36 and 36. Correct.

Therefore, the integer value of x is 45.

**Final Answer**
The integer value of $ x $ is \boxed{45}.

To determine the integer value of $ x $ in the arithmetic sequence $ 3^2, x, 3^4 $, we begin by identifying the numerical values of the given terms:

- $ 3^2 = 9 $
- $ 3^4 = 81 $

So the sequence becomes:
$$
9, \ x, \ 81
$$

---

### Step 1: Understand the properties of an arithmetic sequence

In an arithmetic sequence, the difference between consecutive terms is constant. That is, the difference between the second and first term is the same as the difference between the third and second term:

$$
x - 9 = 81 - x
$$

---

### Step 2: Solve the equation

We solve the equation step by step:

$$
x - 9 = 81 - x
$$

Add $ x $ to both sides:

$$
2x - 9 = 81
$$

Add 9 to both sides:

$$
2x = 90
$$

Divide both sides by 2:

$$
x = 45
$$

---

### Step 3: Verify the result

Let’s check if the sequence $ 9, 45, 81 $ is indeed arithmetic:

- Difference between 45 and 9: $ 45 - 9 = 36 $
- Difference between 81 and 45: $ 81 - 45 = 36 $

Since the common difference is consistent, the sequence is arithmetic, and the value of $ x $ is correct.

---

### Final Answer

$$
\boxed{45}
$$

[Assistant]
Explanation:
Okay, so I need to find the integer value of x in the arithmetic sequence 3², x, 3⁴. Let me start by recalling what an arithmetic sequence is. An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. That means if I have three terms in an arithmetic sequence, the middle term is the average of the first and third terms.

First, let me write down the given terms. The first term is 3 squared, which is 9. The second term is x, which is what we need to find. The third term is 3 to the fourth power, which is 81. So the sequence is 9, x, 81.

Since it's an arithmetic sequence, the difference between the first and second term should be the same as the difference between the second and third term. Let me denote the common difference as d. So, the difference between the first and second term is x - 9, and the difference between the second and third term is 81 - x. Since these differences are equal, I can set up the equation:

x - 9 = 81 - x

Now, I need to solve for x. Let me do that step by step.

First, I can add x to both sides of the equation to get rid of the -x on the right side.

x - 9 + x = 81 - x + x

Simplifying both sides:

2x - 9 = 81

Next, I need to isolate the term with x. So, I'll add 9 to both sides:

2x - 9 + 9 = 81 + 9

Which simplifies to:

2x = 90

Now, to solve for x, I divide both sides by 2:

x = 90 / 2

x = 45

Wait, let me check if that makes sense. If the first term is 9, the second term is 45, and the third term is 81, then the common difference should be 45 - 9 = 36, and 81 - 45 = 36. Yes, that's correct. Both differences are 36, so it is an arithmetic sequence.

But let me make sure I didn't make any mistakes in my calculations. Let me go through the steps again.

Original equation: x - 9 = 81 - x

Adding x to both sides: x + x - 9 = 81

Which is 2x - 9 = 81

Adding 9 to both sides: 2x = 90

Divide by 2: x = 45. Yep, that seems right.

Alternatively, since in an arithmetic sequence the middle term is the average of the first and third terms, I can use the formula for the nth term. For three terms, the second term is the average of the first and third. So:

x = (first term + third term)/2 = (9 + 81)/2 = 90/2 = 45.

Same result. So that confirms it.

Is there another way to approach this problem? Maybe by considering the positions of the terms? Let me think. In an arithmetic sequence, each term is obtained by adding the common difference to the previous term. So, if the first term is a₁ = 9, then the second term a₂ = a₁ + d, and the third term a₃ = a₂ + d = a₁ + 2d.

Given that a₃ is 81, so:

a₁ + 2d = 81

We know a₁ is 9, so:

9 + 2d = 81

Subtract 9 from both sides:

2d = 72

Divide by 2:

d = 36

Then, the second term a₂ = a₁ + d = 9 + 36 = 45.

Same answer again. So, regardless of the method, I get x = 45.

Is there a possibility that the question is referring to something else? For example, maybe the sequence is not arithmetic but geometric? Wait, the question specifically says arithmetic sequence, so no. But just to check, if it were geometric, the ratio between terms would be constant. Let me see what would happen in that case.

In a geometric sequence, the middle term squared would be equal to the product of the first and third terms. So, x² = 9 * 81 = 729. Then x would be sqrt(729) = 27 or -27. But since the question says arithmetic sequence, not geometric, so that's not relevant here.

But just to make sure, the problem says arithmetic sequence, so we can ignore the geometric possibility.

Another check: Let me verify the arithmetic sequence with x = 45. The terms are 9, 45, 81. The difference between 9 and 45 is 36, and between 45 and 81 is also 36. So yes, that's an arithmetic sequence with common difference 36.

Therefore, I think x = 45 is the correct answer.

Wait, but the problem says "the integer value of x". Well, 45 is an integer, so that's fine.

Is there any other integer that could satisfy this? Let me think. Suppose someone thought that maybe the sequence is in some other form? For example, maybe the exponents are in arithmetic sequence? But the problem says the sequence is 3², x, 3⁴. So the terms themselves are 3², x, 3⁴. So the terms are 9, x, 81. So unless there's some other interpretation, but I don't think so.

Alternatively, maybe the exponents are in arithmetic sequence? Let me check. If the exponents were in arithmetic sequence, then 2, x, 4. Then the middle term would be (2 + 4)/2 = 3. So x would be 3, and then the terms would be 3², 3³, 3⁴. But that would be a geometric sequence with common ratio 3, not arithmetic. However, the problem says the sequence is arithmetic, so that's not the case.

Therefore, the original interpretation is correct. The terms themselves are in arithmetic sequence, so x is 45.

Another way to think about it: The difference between the first and third term is 81 - 9 = 72. In an arithmetic sequence, the difference between the first and third term is twice the common difference. So 2d = 72, so d = 36. Then, the second term is first term + d = 9 + 36 = 45.

All methods lead to the same answer. Therefore, I can be confident that x is 45.

Just to recap:

Given arithmetic sequence: 3², x, 3⁴

Convert to numbers: 9, x, 81

Common difference d = x - 9 = 81 - x

Solving gives x = 45

Check: 9, 45, 81 with differences 36 and 36. Correct.

Therefore, the integer value of x is 45.

**Final Answer**
The integer value of $ x $ is \boxed{45}.

To determine the integer value of $ x $ in the arithmetic sequence $ 3^2, x, 3^4 $, we begin by identifying the numerical values of the given terms:

- $ 3^2 = 9 $
- $ 3^4 = 81 $

So the sequence becomes:
$$
9, \ x, \ 81
$$

---

### Step 1: Understand the properties of an arithmetic sequence

In an arithmetic sequence, the difference between consecutive terms is constant. That is, the difference between the second and first term is the same as the difference between the third and second term:

$$
x - 9 = 81 - x
$$

---

### Step 2: Solve the equation

We solve the equation step by step:

$$
x - 9 = 81 - x
$$

Add $ x $ to both sides:

$$
2x - 9 = 81
$$

Add 9 to both sides:

$$
2x = 90
$$

Divide both sides by 2:

$$
x = 45
$$

---

### Step 3: Verify the result

Let’s check if the sequence $ 9, 45, 81 $ is indeed arithmetic:

- Difference between 45 and 9: $ 45 - 9 = 36 $
- Difference between 81 and 45: $ 81 - 45 = 36 $

Since the common difference is consistent, the sequence is arithmetic, and the value of $ x $ is correct.

---

### Final Answer

$$
\boxed{45}
$$

Solution: The term $x$ is simply the average of $3^2 = 9$ and $3^4 = 81$, which is $(9 + 81)/2 = 90/2 = \boxed{45}$.