Explanation:
Okay, so Karla drove her vehicle 12,000 miles last year. The question is asking how many gallons of gasoline she would have saved if she had driven a fuel-efficient hybrid car instead of an SUV. The hybrid averages 48 miles per gallon, and the SUV averages 15 miles per gallon. Alright, let me think through this step by step.

First, I need to figure out how much gasoline she used with the SUV and then compare that to how much she would have used with the hybrid. The difference between those two amounts should be the gallons saved, right?

So, to find out the gasoline consumption for each vehicle, I need to divide the total miles driven by the miles per gallon (mpg) rating of each vehicle. That makes sense because miles per gallon tells you how many miles the car can go on one gallon of gas. So, dividing total miles by mpg will give me the number of gallons needed for that trip.

Let me write that down. For the SUV:

Gallons used by SUV = Total miles / SUV mpg = 12,000 miles / 15 mpg.

And for the hybrid:

Gallons used by hybrid = Total miles / Hybrid mpg = 12,000 miles / 48 mpg.

Then, the gallons saved would be the difference between these two amounts. So:

Gallons saved = Gallons used by SUV - Gallons used by hybrid.

Okay, let me calculate each part.

Starting with the SUV. 12,000 divided by 15. Hmm. Let me do that division. 12,000 divided by 15. Well, 15 times 800 is 12,000 because 15*800 = 12,000. So that means the SUV used 800 gallons.

Wait, let me check that again. 15 times 800: 15*800. 15*8 is 120, so 15*800 is 12,000. Yes, that's correct. So the SUV used 800 gallons.

Now for the hybrid. 12,000 divided by 48. Hmm, that's a bit trickier. Let me see. Maybe I can simplify this fraction. Let's see, both numerator and denominator can be divided by 12? Let me check.

12,000 divided by 12 is 1,000. 48 divided by 12 is 4. So now it becomes 1,000 divided by 4, which is 250. Therefore, 12,000 divided by 48 is 250. So the hybrid would have used 250 gallons.

Therefore, the gallons saved would be 800 minus 250. Let me subtract that. 800 - 250 is 550. So she would have saved 550 gallons.

Wait, let me make sure I didn't make a mistake here. Let me double-check the calculations.

First, for the SUV: 12,000 / 15. Let me think of it as 12,000 divided by 15. 15 goes into 12,000 how many times? Well, 15*1000 = 15,000, which is more than 12,000, so 15*800 = 12,000. Yep, that's right. So 800 gallons.

For the hybrid: 12,000 / 48. Let me do another way. Let's divide numerator and denominator by 6 first. 12,000 / 6 = 2,000. 48 / 6 = 8. So now it's 2,000 / 8 = 250. So same result. So that checks out. So 250 gallons for the hybrid.

Therefore, 800 - 250 is indeed 550. So the answer is 550 gallons saved.

Wait, but let me think again. Is there another way to approach this problem? Maybe using ratios or something else? Let me see.

Alternatively, the difference in mpg between the two cars is 48 - 15 = 33 mpg. But wait, can I just do something with that? Probably not directly, because the gallons saved isn't just total miles divided by the difference in mpg. Because the mpg is a rate, so the gallons used depend on each car's efficiency.

So, for example, if you have two different efficiencies, the total gasoline used is different. So, the way I did it before is correct. It's total miles divided by each mpg to find gallons, then subtract.

Alternatively, if I think about it, the gasoline saved is equal to (total miles) multiplied by (1/SUV mpg - 1/hybrid mpg). Wait, let me check.

Wait, gallons used by SUV is total miles / SUV mpg, and gallons used by hybrid is total miles / hybrid mpg. Therefore, gallons saved is total miles*(1/SUV mpg - 1/hybrid mpg). Wait, no, it's (total miles / SUV mpg) - (total miles / hybrid mpg) = total miles*(1/SUV mpg - 1/hybrid mpg). But since we want the gallons saved, it's actually (total miles / SUV mpg) - (total miles / hybrid mpg). Which would be total miles*(1/SUV mpg - 1/hybrid mpg). But since hybrid is more efficient, 1/hybrid mpg is smaller than 1/SUV mpg, so the difference would be positive. Wait, but in our case, hybrid is more efficient, so gallons saved is SUV gallons minus hybrid gallons. So, yes, that formula would work.

So plugging in numbers:

Total miles = 12,000

1/SUV mpg = 1/15

1/hybrid mpg = 1/48

Difference is 1/15 - 1/48. Let me compute that. To subtract fractions, common denominator. The least common denominator of 15 and 48. Let me see. Prime factors of 15: 3*5. Prime factors of 48: 2^4 * 3. So LCD would be 2^4 * 3 *5 = 16*3*5 = 240. So converting:

1/15 = 16/240

1/48 = 5/240

Therefore, 16/240 - 5/240 = 11/240.

Therefore, gallons saved = 12,000 * (11/240). Let me compute that. 12,000 divided by 240 multiplied by 11.

First, 12,000 divided by 240. Let me simplify. 240 goes into 12,000 how many times? Well, 240*50=12,000. So 12,000 / 240 = 50. Therefore, 50*11=550. So same result. Therefore, 550 gallons. So that confirms my previous answer.

Therefore, regardless of the method, the answer is 550 gallons saved.

Wait, but let me check once more with another approach. Suppose I calculate the fuel efficiency difference in some other way. For instance, if I think about how much more efficient the hybrid is. But I think that might complicate things. Let me try.

Alternatively, think about how many gallons each car uses. For the SUV, 800 gallons. For the hybrid, 250 gallons. So difference is 550. That's straightforward. So, yes.

Alternatively, maybe think about the miles per gallon difference. But as I thought before, that's not straightforward. Because if you have a higher mpg, the difference in mpg doesn't directly translate to gallons saved unless you have a fixed amount of miles. Which we do here, 12,000 miles. So in this case, since miles are fixed, the gallons saved is dependent on how much each car uses.

Alternatively, if you wanted to think in terms of how much more miles the hybrid can go on a gallon, but that might not be helpful here.

Alternatively, think of it as the SUV uses 15 miles per gallon, so for every gallon, you can go 15 miles. The hybrid can go 48 miles per gallon. So, for the same number of miles, how much less gas does the hybrid use? Well, that's exactly what we calculated.

Alternatively, maybe think in terms of fuel economy. The hybrid is 48 mpg, SUV is 15 mpg. So, the hybrid is more efficient. So, to find out how much fuel is saved, you can compute the fuel used by SUV minus fuel used by hybrid. Which again is the same thing.

So, all methods lead to 550 gallons saved. Therefore, I think that is correct.

But to be thorough, let me check my first calculation again. SUV uses 12,000 / 15. Let me compute 12,000 divided by 15. 15 times 800 is 12,000. Yes. So that's 800 gallons. Hybrid is 12,000 divided by 48. Let me do 12,000 divided by 48. 48*200 = 9,600. Subtract that from 12,000: 12,000 - 9,600 = 2,400. Now, 48*50 = 2,400. So 200 + 50 = 250. Therefore, 250 gallons. So difference is 800 - 250 = 550. Correct.

Another way: 48 mpg is 3.2 times better than 15 mpg? Wait, 48 divided by 15 is 3.2. So, the hybrid is 3.2 times more efficient? So, if you have a car that is 3.2 times more efficient, then for the same distance, it uses 1/3.2 times the fuel. So, 1/3.2 is approximately 0.3125. So, 800 * 0.3125 = 250. So, same as before. So 800 - 250 = 550. So that also works. So that's confirming again.

Alternatively, maybe think about the fuel economy ratio. But again, same result.

So, all methods lead to 550 gallons saved. Therefore, I think the answer is 550.

But just to make sure, let me check if there's any trick in the question that I might have missed. The question says "had she driven a fuel-efficient hybrid car averaging 48 miles per gallon rather than an SUV averaging 15 miles per gallon". So, it's comparing the two cars. So, the total miles driven is same, 12,000. So, the difference in fuel consumption is as calculated. Therefore, 550 gallons saved. Seems straightforward.

Is there any chance that the question is asking for something else? Like, maybe the total gallons consumed by both? But no, the question specifically asks "how many gallons of gasoline would she have saved last year had she driven a fuel-efficient hybrid car... rather than an SUV". So, the saved gallons is the difference between the SUV's consumption and the hybrid's consumption. Therefore, 550.

So, I think that's solid. Therefore, the answer is 550 gallons.

**Final Answer**
Karla would have saved \boxed{550} gallons of gasoline last year.

To determine how many gallons of gasoline Karla would have saved by driving a hybrid car instead of an SUV, we begin by analyzing the fuel efficiency of each vehicle and the total distance driven.

---

### Given:
- **Total miles driven last year**: 12,000 miles
- **SUV fuel efficiency**: 15 miles per gallon
- **Hybrid fuel efficiency**: 48 miles per gallon

---

### Step 1: Calculate gallons used by the SUV

To find the gallons of gasoline used by the SUV, divide the total miles by the miles per gallon:

$$
\text{Gallons used by SUV} = \frac{12,000}{15} = 800 \text{ gallons}
$$

---

### Step 2: Calculate gallons used by the hybrid

For the hybrid car, we perform the same calculation:

$$
\text{Gallons used by hybrid} = \frac{12,000}{48} = 250 \text{ gallons}
$$

---

### Step 3: Calculate gallons saved

The difference in fuel consumption is:

$$
\text{Gallons saved} = 800 - 250 = \boxed{550}
$$

---

### Final Answer:
Karla would have saved \boxed{550} gallons of gasoline last year by driving the hybrid car instead of the SUV.

[Assistant]
Explanation:
Okay, so Karla drove her vehicle 12,000 miles last year. The question is asking how many gallons of gasoline she would have saved if she had driven a fuel-efficient hybrid car instead of an SUV. The hybrid averages 48 miles per gallon, and the SUV averages 15 miles per gallon. Alright, let me think through this step by step.

First, I need to figure out how much gasoline she used with the SUV and then compare that to how much she would have used with the hybrid. The difference between those two amounts should be the gallons saved, right?

So, to find out the gasoline consumption for each vehicle, I need to divide the total miles driven by the miles per gallon (mpg) rating of each vehicle. That makes sense because miles per gallon tells you how many miles the car can go on one gallon of gas. So, dividing total miles by mpg will give me the number of gallons needed for that trip.

Let me write that down. For the SUV:

Gallons used by SUV = Total miles / SUV mpg = 12,000 miles / 15 mpg.

And for the hybrid:

Gallons used by hybrid = Total miles / Hybrid mpg = 12,000 miles / 48 mpg.

Then, the gallons saved would be the difference between these two amounts. So:

Gallons saved = Gallons used by SUV - Gallons used by hybrid.

Okay, let me calculate each part.

Starting with the SUV. 12,000 divided by 15. Hmm. Let me do that division. 12,000 divided by 15. Well, 15 times 800 is 12,000 because 15*800 = 12,000. So that means the SUV used 800 gallons.

Wait, let me check that again. 15 times 800: 15*800. 15*8 is 120, so 15*800 is 12,000. Yes, that's correct. So the SUV used 800 gallons.

Now for the hybrid. 12,000 divided by 48. Hmm, that's a bit trickier. Let me see. Maybe I can simplify this fraction. Let's see, both numerator and denominator can be divided by 12? Let me check.

12,000 divided by 12 is 1,000. 48 divided by 12 is 4. So now it becomes 1,000 divided by 4, which is 250. Therefore, 12,000 divided by 48 is 250. So the hybrid would have used 250 gallons.

Therefore, the gallons saved would be 800 minus 250. Let me subtract that. 800 - 250 is 550. So she would have saved 550 gallons.

Wait, let me make sure I didn't make a mistake here. Let me double-check the calculations.

First, for the SUV: 12,000 / 15. Let me think of it as 12,000 divided by 15. 15 goes into 12,000 how many times? Well, 15*1000 = 15,000, which is more than 12,000, so 15*800 = 12,000. Yep, that's right. So 800 gallons.

For the hybrid: 12,000 / 48. Let me do another way. Let's divide numerator and denominator by 6 first. 12,000 / 6 = 2,000. 48 / 6 = 8. So now it's 2,000 / 8 = 250. So same result. So that checks out. So 250 gallons for the hybrid.

Therefore, 800 - 250 is indeed 550. So the answer is 550 gallons saved.

Wait, but let me think again. Is there another way to approach this problem? Maybe using ratios or something else? Let me see.

Alternatively, the difference in mpg between the two cars is 48 - 15 = 33 mpg. But wait, can I just do something with that? Probably not directly, because the gallons saved isn't just total miles divided by the difference in mpg. Because the mpg is a rate, so the gallons used depend on each car's efficiency.

So, for example, if you have two different efficiencies, the total gasoline used is different. So, the way I did it before is correct. It's total miles divided by each mpg to find gallons, then subtract.

Alternatively, if I think about it, the gasoline saved is equal to (total miles) multiplied by (1/SUV mpg - 1/hybrid mpg). Wait, let me check.

Wait, gallons used by SUV is total miles / SUV mpg, and gallons used by hybrid is total miles / hybrid mpg. Therefore, gallons saved is total miles*(1/SUV mpg - 1/hybrid mpg). Wait, no, it's (total miles / SUV mpg) - (total miles / hybrid mpg) = total miles*(1/SUV mpg - 1/hybrid mpg). But since we want the gallons saved, it's actually (total miles / SUV mpg) - (total miles / hybrid mpg). Which would be total miles*(1/SUV mpg - 1/hybrid mpg). But since hybrid is more efficient, 1/hybrid mpg is smaller than 1/SUV mpg, so the difference would be positive. Wait, but in our case, hybrid is more efficient, so gallons saved is SUV gallons minus hybrid gallons. So, yes, that formula would work.

So plugging in numbers:

Total miles = 12,000

1/SUV mpg = 1/15

1/hybrid mpg = 1/48

Difference is 1/15 - 1/48. Let me compute that. To subtract fractions, common denominator. The least common denominator of 15 and 48. Let me see. Prime factors of 15: 3*5. Prime factors of 48: 2^4 * 3. So LCD would be 2^4 * 3 *5 = 16*3*5 = 240. So converting:

1/15 = 16/240

1/48 = 5/240

Therefore, 16/240 - 5/240 = 11/240.

Therefore, gallons saved = 12,000 * (11/240). Let me compute that. 12,000 divided by 240 multiplied by 11.

First, 12,000 divided by 240. Let me simplify. 240 goes into 12,000 how many times? Well, 240*50=12,000. So 12,000 / 240 = 50. Therefore, 50*11=550. So same result. Therefore, 550 gallons. So that confirms my previous answer.

Therefore, regardless of the method, the answer is 550 gallons saved.

Wait, but let me check once more with another approach. Suppose I calculate the fuel efficiency difference in some other way. For instance, if I think about how much more efficient the hybrid is. But I think that might complicate things. Let me try.

Alternatively, think about how many gallons each car uses. For the SUV, 800 gallons. For the hybrid, 250 gallons. So difference is 550. That's straightforward. So, yes.

Alternatively, maybe think about the miles per gallon difference. But as I thought before, that's not straightforward. Because if you have a higher mpg, the difference in mpg doesn't directly translate to gallons saved unless you have a fixed amount of miles. Which we do here, 12,000 miles. So in this case, since miles are fixed, the gallons saved is dependent on how much each car uses.

Alternatively, if you wanted to think in terms of how much more miles the hybrid can go on a gallon, but that might not be helpful here.

Alternatively, think of it as the SUV uses 15 miles per gallon, so for every gallon, you can go 15 miles. The hybrid can go 48 miles per gallon. So, for the same number of miles, how much less gas does the hybrid use? Well, that's exactly what we calculated.

Alternatively, maybe think in terms of fuel economy. The hybrid is 48 mpg, SUV is 15 mpg. So, the hybrid is more efficient. So, to find out how much fuel is saved, you can compute the fuel used by SUV minus fuel used by hybrid. Which again is the same thing.

So, all methods lead to 550 gallons saved. Therefore, I think that is correct.

But to be thorough, let me check my first calculation again. SUV uses 12,000 / 15. Let me compute 12,000 divided by 15. 15 times 800 is 12,000. Yes. So that's 800 gallons. Hybrid is 12,000 divided by 48. Let me do 12,000 divided by 48. 48*200 = 9,600. Subtract that from 12,000: 12,000 - 9,600 = 2,400. Now, 48*50 = 2,400. So 200 + 50 = 250. Therefore, 250 gallons. So difference is 800 - 250 = 550. Correct.

Another way: 48 mpg is 3.2 times better than 15 mpg? Wait, 48 divided by 15 is 3.2. So, the hybrid is 3.2 times more efficient? So, if you have a car that is 3.2 times more efficient, then for the same distance, it uses 1/3.2 times the fuel. So, 1/3.2 is approximately 0.3125. So, 800 * 0.3125 = 250. So, same as before. So 800 - 250 = 550. So that also works. So that's confirming again.

Alternatively, maybe think about the fuel economy ratio. But again, same result.

So, all methods lead to 550 gallons saved. Therefore, I think the answer is 550.

But just to make sure, let me check if there's any trick in the question that I might have missed. The question says "had she driven a fuel-efficient hybrid car averaging 48 miles per gallon rather than an SUV averaging 15 miles per gallon". So, it's comparing the two cars. So, the total miles driven is same, 12,000. So, the difference in fuel consumption is as calculated. Therefore, 550 gallons saved. Seems straightforward.

Is there any chance that the question is asking for something else? Like, maybe the total gallons consumed by both? But no, the question specifically asks "how many gallons of gasoline would she have saved last year had she driven a fuel-efficient hybrid car... rather than an SUV". So, the saved gallons is the difference between the SUV's consumption and the hybrid's consumption. Therefore, 550.

So, I think that's solid. Therefore, the answer is 550 gallons.

**Final Answer**
Karla would have saved \boxed{550} gallons of gasoline last year.

To determine how many gallons of gasoline Karla would have saved by driving a hybrid car instead of an SUV, we begin by analyzing the fuel efficiency of each vehicle and the total distance driven.

---

### Given:
- **Total miles driven last year**: 12,000 miles
- **SUV fuel efficiency**: 15 miles per gallon
- **Hybrid fuel efficiency**: 48 miles per gallon

---

### Step 1: Calculate gallons used by the SUV

To find the gallons of gasoline used by the SUV, divide the total miles by the miles per gallon:

$$
\text{Gallons used by SUV} = \frac{12,000}{15} = 800 \text{ gallons}
$$

---

### Step 2: Calculate gallons used by the hybrid

For the hybrid car, we perform the same calculation:

$$
\text{Gallons used by hybrid} = \frac{12,000}{48} = 250 \text{ gallons}
$$

---

### Step 3: Calculate gallons saved

The difference in fuel consumption is:

$$
\text{Gallons saved} = 800 - 250 = \boxed{550}
$$

---

### Final Answer:
Karla would have saved \boxed{550} gallons of gasoline last year by driving the hybrid car instead of the SUV.

Solution: Karla used $12000/15 = 800$ gallons to drive 12000 miles. Had she driven the hybrid, she would have used $12000/48 = 250$ gallons. Therefore, she would have saved $800 - 250 = \boxed{550}$ gallons.


Explanation:
Okay, so I need to find the square root of f(f(f(f(1)))), where f(x) is 2 to the power of x. Let me break this down step by step.

First, let me make sure I understand the problem. The function f(x) is defined as 2 raised to the x. So, f(x) = 2^x. Then, the problem is asking for the square root of f applied four times to 1. That is, starting with 1, apply f four times, and then take the square root of the result.

Let me write this out in mathematical terms. The expression we need to compute is √[f(f(f(f(1)))].

Alright, let me start by computing the innermost function first. Let's start with f(1).

So, f(1) = 2^1. Well, 2^1 is 2. So that's straightforward.

Now, moving one step out, we need to compute f(f(1)), which is f(2). Since f(2) = 2^2, which is 4.

Next, we need to compute f(f(f(1))) which is f(4). Let me check that again. Wait, the innermost was f(1) = 2, then f(f(1)) is f(2) = 4, then f(f(f(1))) would be f(4). So yes, f(4) = 2^4. 2^4 is 16.

Continuing on, the next layer is f(f(f(f(1)))) which is f(16). So f(16) is 2^16.

Therefore, the entire expression inside the square root is 2^16. So now, we need to compute the square root of 2^16.

Let me recall that the square root of a number is the same as raising that number to the power of 1/2. Therefore, √(2^16) = (2^16)^(1/2).

When you raise a power to another power, you multiply the exponents. So 16 * (1/2) is 8. Therefore, this simplifies to 2^8.

Now, 2^8 is 256. Let me verify that. 2^10 is 1024, so 2^8 is 2^10 divided by 2^2, which is 1024 / 4 = 256. Yes, that seems right.

So putting it all together: starting with 1, applying f four times gives us 2^16, then taking the square root gives us 2^8, which is 256.

Let me just check each step again to make sure I didn't make a mistake.

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

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

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

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

Then square root of 2^16: sqrt(2^16) = 2^(16/2) = 2^8 = 256. Correct.

Hmm, seems straightforward. But maybe I should verify the exponents again. Let me think.

Each time we apply f, we're taking the previous result and raising 2 to that power. So starting with 1, then 2^1 = 2. Then 2^2 = 4. Then 2^4 = 16. Then 2^16. So yes, that's 2^(2^(2^(2^1)))? Wait, no, hold on. Wait, is that the case?

Wait, hold on, perhaps there's confusion here. Let me think again.

Wait, f(x) = 2^x. So f(f(f(f(1)))) is f(f(f(2^1))) which is f(f(2^2)) which is f(2^4) which is 2^(2^4) which is 2^16. So yes, so that's 2^16. Then square root of that is 2^8.

Alternatively, maybe I can think of it as exponents. Let me try to represent each step:

Let me denote f^n(x) as applying f n times. So f^1(x) = f(x) = 2^x. Then f^2(x) = f(f(x)) = 2^(2^x). Wait, is that correct? Wait, no. Let me clarify.

Wait, f(x) is 2^x. Therefore, f(f(x)) is f(2^x) = 2^(2^x). So yes, f^2(x) = 2^(2^x). Then f^3(x) = f(f^2(x)) = 2^(f^2(x)) = 2^(2^(2^x)). Similarly, f^4(x) = 2^(2^(2^(2^x))).

Therefore, if we plug in x = 1, then f^4(1) = 2^(2^(2^(2^1))) = 2^(2^(2^2)) = 2^(2^4) = 2^16. Which matches what I had before. Then sqrt(f^4(1)) = sqrt(2^16) = 2^8 = 256.

Therefore, the answer is 256.

But let me check again with another approach. Maybe by calculating each step numerically:

Start with 1.

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

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

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

Fourth f(f(f(f(1)))) = f(16) = 2^16 = 65536.

Then sqrt(65536) = 256.

Yes, because 256 squared is 65536. Let me check that. 256 * 256: 256*200 = 51,200; 256*50 = 12,800; 256*6 = 1,536. So adding those: 51,200 + 12,800 = 64,000; 64,000 + 1,536 = 65,536. Correct.

So sqrt(65536) is indeed 256.

Therefore, the answer is 256.

Wait, but the problem says "Find sqrt(f(f(f(f(1)))))." So that's the square root of f(f(f(f(1)))), which is sqrt(2^16) = 2^8 = 256.

So I think that's solid.

But maybe I can think of another way to approach this. For example, using exponents properties. Let's see:

Each time we apply f, we are exponentiating 2 to the previous result. So starting with 1:

After 1st application: 2^1

After 2nd application: 2^(2^1)

After 3rd application: 2^(2^(2^1))

After 4th application: 2^(2^(2^(2^1)))

So that's 2^(2^(2^2)) = 2^(2^4) = 2^16.

So same as before.

Therefore, taking the square root gives us 2^(16/2) = 2^8 = 256.

So, regardless of the approach, I get 256.

Therefore, I think the answer is 256.

Is there a possibility that I misread the problem? Let me check again.

The problem says: Let f(x) = 2^x. Find sqrt(f(f(f(f(1))))).

Yes. So four compositions of f, starting with 1, then square root.

Alternatively, maybe the problem is written with parentheses differently? Like sqrt(f(f(f(f(1))))) or maybe sqrt(f(f(f(f(1)))))? But I think the way it's written is clear.

Alternatively, maybe someone could misinterpret the square root as being applied after each function, but the problem says sqrt(f(f(f(f(1))))), so it's the square root of the entire composition.

Alternatively, maybe the problem is written with different parentheses, but the original says: sqrt{f(f(f(f(1))))}

So, yeah, the square root is only over the entire f(f(f(f(1)))).

So I think my reasoning is correct.

Therefore, the answer is 256.

**Final Answer**
\boxed{256}

To solve the expression
$$
\sqrt{f(f(f(f(1))))}
$$
where $ f(x) = 2^x $, we follow a clear, step-by-step evaluation.

---

### Step 1: Evaluate the innermost function

We start with the innermost function:
$$
f(1) = 2^1 = 2
$$

---

### Step 2: Apply function again

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

---

### Step 3: Apply function again

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

---

### Step 4: Apply function one more time

$$
f(f(f(f(1)))) = f(16) = 2^{16}
$$

---

### Step 5: Take the square root

Now, we compute the square root of the result:
$$
\sqrt{2^{16}} = (2^{16})^{1/2} = 2^{8}
$$
because when raising a power to another power, we multiply the exponents.

$$
2^8 = 256
$$

---

### 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 to the power of x. Let me break this down step by step.

First, let me make sure I understand the problem. The function f(x) is defined as 2 raised to the x. So, f(x) = 2^x. Then, the problem is asking for the square root of f applied four times to 1. That is, starting with 1, apply f four times, and then take the square root of the result.

Let me write this out in mathematical terms. The expression we need to compute is √[f(f(f(f(1)))].

Alright, let me start by computing the innermost function first. Let's start with f(1).

So, f(1) = 2^1. Well, 2^1 is 2. So that's straightforward.

Now, moving one step out, we need to compute f(f(1)), which is f(2). Since f(2) = 2^2, which is 4.

Next, we need to compute f(f(f(1))) which is f(4). Let me check that again. Wait, the innermost was f(1) = 2, then f(f(1)) is f(2) = 4, then f(f(f(1))) would be f(4). So yes, f(4) = 2^4. 2^4 is 16.

Continuing on, the next layer is f(f(f(f(1)))) which is f(16). So f(16) is 2^16.

Therefore, the entire expression inside the square root is 2^16. So now, we need to compute the square root of 2^16.

Let me recall that the square root of a number is the same as raising that number to the power of 1/2. Therefore, √(2^16) = (2^16)^(1/2).

When you raise a power to another power, you multiply the exponents. So 16 * (1/2) is 8. Therefore, this simplifies to 2^8.

Now, 2^8 is 256. Let me verify that. 2^10 is 1024, so 2^8 is 2^10 divided by 2^2, which is 1024 / 4 = 256. Yes, that seems right.

So putting it all together: starting with 1, applying f four times gives us 2^16, then taking the square root gives us 2^8, which is 256.

Let me just check each step again to make sure I didn't make a mistake.

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

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

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

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

Then square root of 2^16: sqrt(2^16) = 2^(16/2) = 2^8 = 256. Correct.

Hmm, seems straightforward. But maybe I should verify the exponents again. Let me think.

Each time we apply f, we're taking the previous result and raising 2 to that power. So starting with 1, then 2^1 = 2. Then 2^2 = 4. Then 2^4 = 16. Then 2^16. So yes, that's 2^(2^(2^(2^1)))? Wait, no, hold on. Wait, is that the case?

Wait, hold on, perhaps there's confusion here. Let me think again.

Wait, f(x) = 2^x. So f(f(f(f(1)))) is f(f(f(2^1))) which is f(f(2^2)) which is f(2^4) which is 2^(2^4) which is 2^16. So yes, so that's 2^16. Then square root of that is 2^8.

Alternatively, maybe I can think of it as exponents. Let me try to represent each step:

Let me denote f^n(x) as applying f n times. So f^1(x) = f(x) = 2^x. Then f^2(x) = f(f(x)) = 2^(2^x). Wait, is that correct? Wait, no. Let me clarify.

Wait, f(x) is 2^x. Therefore, f(f(x)) is f(2^x) = 2^(2^x). So yes, f^2(x) = 2^(2^x). Then f^3(x) = f(f^2(x)) = 2^(f^2(x)) = 2^(2^(2^x)). Similarly, f^4(x) = 2^(2^(2^(2^x))).

Therefore, if we plug in x = 1, then f^4(1) = 2^(2^(2^(2^1))) = 2^(2^(2^2)) = 2^(2^4) = 2^16. Which matches what I had before. Then sqrt(f^4(1)) = sqrt(2^16) = 2^8 = 256.

Therefore, the answer is 256.

But let me check again with another approach. Maybe by calculating each step numerically:

Start with 1.

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

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

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

Fourth f(f(f(f(1)))) = f(16) = 2^16 = 65536.

Then sqrt(65536) = 256.

Yes, because 256 squared is 65536. Let me check that. 256 * 256: 256*200 = 51,200; 256*50 = 12,800; 256*6 = 1,536. So adding those: 51,200 + 12,800 = 64,000; 64,000 + 1,536 = 65,536. Correct.

So sqrt(65536) is indeed 256.

Therefore, the answer is 256.

Wait, but the problem says "Find sqrt(f(f(f(f(1)))))." So that's the square root of f(f(f(f(1)))), which is sqrt(2^16) = 2^8 = 256.

So I think that's solid.

But maybe I can think of another way to approach this. For example, using exponents properties. Let's see:

Each time we apply f, we are exponentiating 2 to the previous result. So starting with 1:

After 1st application: 2^1

After 2nd application: 2^(2^1)

After 3rd application: 2^(2^(2^1))

After 4th application: 2^(2^(2^(2^1)))

So that's 2^(2^(2^2)) = 2^(2^4) = 2^16.

So same as before.

Therefore, taking the square root gives us 2^(16/2) = 2^8 = 256.

So, regardless of the approach, I get 256.

Therefore, I think the answer is 256.

Is there a possibility that I misread the problem? Let me check again.

The problem says: Let f(x) = 2^x. Find sqrt(f(f(f(f(1))))).

Yes. So four compositions of f, starting with 1, then square root.

Alternatively, maybe the problem is written with parentheses differently? Like sqrt(f(f(f(f(1))))) or maybe sqrt(f(f(f(f(1)))))? But I think the way it's written is clear.

Alternatively, maybe someone could misinterpret the square root as being applied after each function, but the problem says sqrt(f(f(f(f(1))))), so it's the square root of the entire composition.

Alternatively, maybe the problem is written with different parentheses, but the original says: sqrt{f(f(f(f(1))))}

So, yeah, the square root is only over the entire f(f(f(f(1)))).

So I think my reasoning is correct.

Therefore, the answer is 256.

**Final Answer**
\boxed{256}

To solve the expression
$$
\sqrt{f(f(f(f(1))))}
$$
where $ f(x) = 2^x $, we follow a clear, step-by-step evaluation.

---

### Step 1: Evaluate the innermost function

We start with the innermost function:
$$
f(1) = 2^1 = 2
$$

---

### Step 2: Apply function again

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

---

### Step 3: Apply function again

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

---

### Step 4: Apply function one more time

$$
f(f(f(f(1)))) = f(16) = 2^{16}
$$

---

### Step 5: Take the square root

Now, we compute the square root of the result:
$$
\sqrt{2^{16}} = (2^{16})^{1/2} = 2^{8}
$$
because when raising a power to another power, we multiply the exponents.

$$
2^8 = 256
$$

---

### 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. Hmm, arithmetic sequence... right, that means the difference between consecutive terms is constant. Let me recall: in an arithmetic sequence, each term after the first is obtained by adding a common difference. So if I have three terms: a, b, c, then b - a = c - b, which means 2b = a + c. That seems right.

So applying that to this problem, the three terms are 3², x, 3⁴. Let me write them out numerically. First term is 3 squared, which is 9. The second term is x, which we need to find. The third term is 3 to the fourth power, which is 81. So the sequence is 9, x, 81.

So according to the arithmetic sequence property, the difference between the first and second term should be the same as the difference between the second and third term. So, x - 9 should equal 81 - x. Let me write that equation down:

x - 9 = 81 - x

Now, solving for x. Let me add x to both sides to get:

x + x - 9 = 81

Which simplifies to:

2x - 9 = 81

Then, add 9 to both sides:

2x = 81 + 9

Which is:

2x = 90

Then divide both sides by 2:

x = 45

Wait, so is x equal to 45? Let me check if that makes sense. If the sequence is 9, 45, 81, then the difference between the first and second term is 45 - 9 = 36. The difference between the second and third term is 81 - 45 = 36. Yep, that's the same difference. So that works.

But let me make sure I didn't make a mistake with the exponents. The first term is 3 squared, which is 9. The third term is 3 to the fourth power, which is 3*3*3*3. Let me compute that again. 3*3 is 9, 9*3 is 27, 27*3 is 81. Yep, that's right. So 3⁴ is 81. So the terms are 9, x, 81.

So solving the equation again: x is the average of the first and third term. Because in an arithmetic sequence, the middle term is the average of the terms around it. So x = (9 + 81)/2. Let me compute that. 9 + 81 is 90, divided by 2 is 45. Yep, so that gives me 45 again.

Therefore, the integer value of x is 45. Wait, but just to make sure there's no trick here. The problem says "arithmetic sequence", so it's not a geometric sequence or anything else. So the difference is constant. So 9, 45, 81. The difference is 36 each time.

Alternatively, if I think of the general formula for an arithmetic sequence: a_n = a_1 + (n-1)d. So here, the first term a_1 is 9, the second term a_2 is x, and the third term a_3 is 81.

So for the second term: x = 9 + (2-1)d = 9 + d.

For the third term: 81 = 9 + (3-1)d = 9 + 2d.

So from the third term equation: 81 = 9 + 2d. Subtract 9: 72 = 2d. Therefore, d = 36. Then plugging back into the second term equation: x = 9 + 36 = 45. Same answer.

So all methods lead me to x = 45. Therefore, the integer value is 45.

But just to be thorough, let me check if maybe the problem was expecting something else. For example, could there be another interpretation? Maybe the sequence is written as 3², x, 3⁴, but is there a different way to interpret the positions? Like, maybe the sequence is in exponents? Wait, but the problem says "arithmetic sequence", so it's about the values, not the exponents. So unless there's a trick where the exponents form an arithmetic sequence, but that would be a different problem.

For example, if the problem was about exponents, like 3², x, 3⁴ being exponents in an arithmetic sequence, but that's not what the problem says. The problem says "the arithmetic sequence 3², x, 3⁴", so the terms are 3², x, 3⁴, which are numbers, not exponents. So yeah, 9, x, 81.

Alternatively, maybe the problem is in another base? But no, the problem doesn't mention anything about bases, so we can assume base 10.

Another check: maybe the question is written as 3^2, x, 3^4, but maybe someone might misread it as 3^2, x^2, 3^4? But no, the original problem says "arithmetic sequence 3², x, 3⁴". So the terms are 3 squared, then x, then 3 to the fourth.

So, no, that's not the case. So x is just the middle term.

Therefore, I think I can be confident that x is 45.

Wait, but let me check once more with the formula. The common difference is 36, so starting at 9, adding 36 gives 45, adding another 36 gives 81. Perfect.

Alternatively, maybe if I consider that arithmetic sequences can have negative differences? But in this case, since we are going from 9 to 81, the difference has to be positive, so x has to be larger than 9, which 45 is.

Alternatively, if someone thought that maybe the sequence is decreasing? But 3^4 is larger than 3^2, so the sequence is increasing. Therefore, the common difference is positive.

Another check: suppose I thought that maybe x is between 3² and 3^4 in exponents? Like, if the exponents formed an arithmetic sequence. For example, if the exponents were 2, x, 4, then x would be 3, but that would be if the exponents are in arithmetic sequence. However, the problem says the sequence is 3², x, 3⁴, not 2, x, 4. Therefore, that would be a different problem.

But given the problem is stated as it is, I think the correct interpretation is that the terms are 3², x, 3⁴ as numbers. Therefore, x is 45.

So I think that's solid. Therefore, the integer value is 45.

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

To solve the problem, we are given an arithmetic sequence:
$$
3^2,\ x,\ 3^4
$$

---

### Step 1: Evaluate the known terms

We convert the exponential terms into their numerical values:

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

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

---

### Step 2: Apply the property of an arithmetic sequence

In an arithmetic sequence, the difference between consecutive terms is constant. This implies:

$$
x - 9 = 81 - x
$$

---

### Step 3: Solve the equation

Start by adding $x$ to both sides:

$$
x + x - 9 = 81 \Rightarrow 2x - 9 = 81
$$

Then add 9 to both sides:

$$
2x = 90
$$

Now, divide by 2:

$$
x = 45
$$

---

### Step 4: Verify the solution

Let’s check if the sequence $9,\ 45,\ 81$ forms an arithmetic sequence:

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

Both differences are equal, confirming our solution.

---

### Final Answer

$$
\boxed{45}
$$

[Assistant]
Explanation:
Okay, so I need to find the integer value of x in the arithmetic sequence 3², x, 3⁴. Hmm, arithmetic sequence... right, that means the difference between consecutive terms is constant. Let me recall: in an arithmetic sequence, each term after the first is obtained by adding a common difference. So if I have three terms: a, b, c, then b - a = c - b, which means 2b = a + c. That seems right.

So applying that to this problem, the three terms are 3², x, 3⁴. Let me write them out numerically. First term is 3 squared, which is 9. The second term is x, which we need to find. The third term is 3 to the fourth power, which is 81. So the sequence is 9, x, 81.

So according to the arithmetic sequence property, the difference between the first and second term should be the same as the difference between the second and third term. So, x - 9 should equal 81 - x. Let me write that equation down:

x - 9 = 81 - x

Now, solving for x. Let me add x to both sides to get:

x + x - 9 = 81

Which simplifies to:

2x - 9 = 81

Then, add 9 to both sides:

2x = 81 + 9

Which is:

2x = 90

Then divide both sides by 2:

x = 45

Wait, so is x equal to 45? Let me check if that makes sense. If the sequence is 9, 45, 81, then the difference between the first and second term is 45 - 9 = 36. The difference between the second and third term is 81 - 45 = 36. Yep, that's the same difference. So that works.

But let me make sure I didn't make a mistake with the exponents. The first term is 3 squared, which is 9. The third term is 3 to the fourth power, which is 3*3*3*3. Let me compute that again. 3*3 is 9, 9*3 is 27, 27*3 is 81. Yep, that's right. So 3⁴ is 81. So the terms are 9, x, 81.

So solving the equation again: x is the average of the first and third term. Because in an arithmetic sequence, the middle term is the average of the terms around it. So x = (9 + 81)/2. Let me compute that. 9 + 81 is 90, divided by 2 is 45. Yep, so that gives me 45 again.

Therefore, the integer value of x is 45. Wait, but just to make sure there's no trick here. The problem says "arithmetic sequence", so it's not a geometric sequence or anything else. So the difference is constant. So 9, 45, 81. The difference is 36 each time.

Alternatively, if I think of the general formula for an arithmetic sequence: a_n = a_1 + (n-1)d. So here, the first term a_1 is 9, the second term a_2 is x, and the third term a_3 is 81.

So for the second term: x = 9 + (2-1)d = 9 + d.

For the third term: 81 = 9 + (3-1)d = 9 + 2d.

So from the third term equation: 81 = 9 + 2d. Subtract 9: 72 = 2d. Therefore, d = 36. Then plugging back into the second term equation: x = 9 + 36 = 45. Same answer.

So all methods lead me to x = 45. Therefore, the integer value is 45.

But just to be thorough, let me check if maybe the problem was expecting something else. For example, could there be another interpretation? Maybe the sequence is written as 3², x, 3⁴, but is there a different way to interpret the positions? Like, maybe the sequence is in exponents? Wait, but the problem says "arithmetic sequence", so it's about the values, not the exponents. So unless there's a trick where the exponents form an arithmetic sequence, but that would be a different problem.

For example, if the problem was about exponents, like 3², x, 3⁴ being exponents in an arithmetic sequence, but that's not what the problem says. The problem says "the arithmetic sequence 3², x, 3⁴", so the terms are 3², x, 3⁴, which are numbers, not exponents. So yeah, 9, x, 81.

Alternatively, maybe the problem is in another base? But no, the problem doesn't mention anything about bases, so we can assume base 10.

Another check: maybe the question is written as 3^2, x, 3^4, but maybe someone might misread it as 3^2, x^2, 3^4? But no, the original problem says "arithmetic sequence 3², x, 3⁴". So the terms are 3 squared, then x, then 3 to the fourth.

So, no, that's not the case. So x is just the middle term.

Therefore, I think I can be confident that x is 45.

Wait, but let me check once more with the formula. The common difference is 36, so starting at 9, adding 36 gives 45, adding another 36 gives 81. Perfect.

Alternatively, maybe if I consider that arithmetic sequences can have negative differences? But in this case, since we are going from 9 to 81, the difference has to be positive, so x has to be larger than 9, which 45 is.

Alternatively, if someone thought that maybe the sequence is decreasing? But 3^4 is larger than 3^2, so the sequence is increasing. Therefore, the common difference is positive.

Another check: suppose I thought that maybe x is between 3² and 3^4 in exponents? Like, if the exponents formed an arithmetic sequence. For example, if the exponents were 2, x, 4, then x would be 3, but that would be if the exponents are in arithmetic sequence. However, the problem says the sequence is 3², x, 3⁴, not 2, x, 4. Therefore, that would be a different problem.

But given the problem is stated as it is, I think the correct interpretation is that the terms are 3², x, 3⁴ as numbers. Therefore, x is 45.

So I think that's solid. Therefore, the integer value is 45.

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

To solve the problem, we are given an arithmetic sequence:
$$
3^2,\ x,\ 3^4
$$

---

### Step 1: Evaluate the known terms

We convert the exponential terms into their numerical values:

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

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

---

### Step 2: Apply the property of an arithmetic sequence

In an arithmetic sequence, the difference between consecutive terms is constant. This implies:

$$
x - 9 = 81 - x
$$

---

### Step 3: Solve the equation

Start by adding $x$ to both sides:

$$
x + x - 9 = 81 \Rightarrow 2x - 9 = 81
$$

Then add 9 to both sides:

$$
2x = 90
$$

Now, divide by 2:

$$
x = 45
$$

---

### Step 4: Verify the solution

Let’s check if the sequence $9,\ 45,\ 81$ forms an arithmetic sequence:

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

Both differences are equal, confirming our solution.

---

### Final Answer

$$
\boxed{45}
$$

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