Explanation:
Okay, let me try to figure out this problem. So, the question is: If a snack-size tin of peaches has 40 calories and that's 2% of a person's daily caloric requirement, how many calories fulfill a person's daily caloric requirement?

Hmm, let me think. So, they're telling me that 40 calories is 2% of the total daily calories someone needs. I need to find out what the total daily caloric requirement is.

Alright, percentages. So, percentage means per hundred. So, 2% is like 2 per 100, right? So, if 40 calories is 2%, then I need to figure out what number (let's call it X) is such that 2% of X is 40 calories.

So, mathematically, that would be 2% of X equals 40. To write that as an equation, it would be 0.02 * X = 40. Because 2% is the same as 0.02 in decimal form.

So, if I have 0.02 * X = 40, I can solve for X by dividing both sides by 0.02. Let me do that.

X = 40 / 0.02. Hmm, let me calculate that. Dividing by 0.02 is the same as multiplying by 50, right? Because 1 divided by 0.02 is 50. So, 40 * 50. Let me compute that.

40 * 50. Well, 40 * 50 is 2000. Wait, so does that mean the total daily caloric requirement is 2000 calories?

Wait, let me check my steps again to make sure I didn't make a mistake.

Starting over: 2% of the daily calories is 40. So, 2% = 0.02. So, 0.02 * daily calories = 40. Therefore, daily calories = 40 / 0.02.

Yes, that seems right. And 40 divided by 0.02. Let me think of another way to compute that. 0.02 goes into 40 how many times?

Well, 0.02 times 1000 is 20. So, 0.02 * 1000 = 20. So, 0.02 * 2000 = 40. Because 20 * 2 = 40. So, yeah, 0.02 * 2000 = 40. Therefore, the daily caloric requirement is 2000 calories.

Alternatively, maybe I can think in terms of proportions. If 2% corresponds to 40 calories, then 1% would correspond to 40 divided by 2, which is 20 calories. Then, 100% would be 20 calories times 100, which is 2000 calories.

Yeah, that method also gives me 2000. So, both ways, I get the same answer.

So, I think that's it. The daily caloric requirement is 2000 calories.

But wait, let me check with another approach. Let's say the total calories are X. So, 2% of X is 40. So, 2/100 * X = 40. So, (2X)/100 = 40. Multiply both sides by 100: 2X = 4000. Divide both sides by 2: X = 2000.

Yep, same answer.

I think that's solid. So, no matter which method I use, I get 2000. So, I think that's the correct answer.

But just to be thorough, let me think if there's any chance that the question is trying to trick me. For example, does it mention anything about the tin being a snack-size, so maybe there's some other consideration? Like, maybe the 40 calories is not the total? But no, the problem says the snack-size tin has 40 calories and that's 2% of the daily requirement. So, the tin's caloric content is 2% of the daily requirement. So, that's straightforward.

No, I don't think there's any trick here. It's a simple percentage problem.

So, conclusion: daily caloric requirement is 2000 calories.

**Final Answer**
The person's daily caloric requirement is \boxed{2000} calories.

To determine a person's daily caloric requirement based on the given information, we begin by understanding that:

> A snack-size tin of peaches contains **40 calories**, and that is **2%** of a person's **daily caloric requirement**.

---

### Step 1: Translate the percentage into a decimal

We know that 2% of the daily requirement is 40 calories. To convert a percentage into its decimal form:

$$
2\% = 0.02
$$

Let the total daily caloric requirement be $ X $. Then, the statement becomes a simple equation:

$$
0.02 \times X = 40
$$

---

### Step 2: Solve the equation

We solve for $ X $ by dividing both sides of the equation by 0.02:

$$
X = \frac{40}{0.02}
$$

$$
X = 2000
$$

---

### Step 3: Verification (Optional, for thoroughness)

Let’s verify by checking that 2% of 2000 equals 40:

$$
0.02 \times 2000 = 40
$$

This confirms the calculation is correct.

---

### Conclusion

Based on the calculation and verification, the person's daily caloric requirement is:

$$
\boxed{2000}
$$

[Assistant]
Explanation:
Okay, let me try to figure out this problem. So, the question is: If a snack-size tin of peaches has 40 calories and that's 2% of a person's daily caloric requirement, how many calories fulfill a person's daily caloric requirement?

Hmm, let me think. So, they're telling me that 40 calories is 2% of the total daily calories someone needs. I need to find out what the total daily caloric requirement is.

Alright, percentages. So, percentage means per hundred. So, 2% is like 2 per 100, right? So, if 40 calories is 2%, then I need to figure out what number (let's call it X) is such that 2% of X is 40 calories.

So, mathematically, that would be 2% of X equals 40. To write that as an equation, it would be 0.02 * X = 40. Because 2% is the same as 0.02 in decimal form.

So, if I have 0.02 * X = 40, I can solve for X by dividing both sides by 0.02. Let me do that.

X = 40 / 0.02. Hmm, let me calculate that. Dividing by 0.02 is the same as multiplying by 50, right? Because 1 divided by 0.02 is 50. So, 40 * 50. Let me compute that.

40 * 50. Well, 40 * 50 is 2000. Wait, so does that mean the total daily caloric requirement is 2000 calories?

Wait, let me check my steps again to make sure I didn't make a mistake.

Starting over: 2% of the daily calories is 40. So, 2% = 0.02. So, 0.02 * daily calories = 40. Therefore, daily calories = 40 / 0.02.

Yes, that seems right. And 40 divided by 0.02. Let me think of another way to compute that. 0.02 goes into 40 how many times?

Well, 0.02 times 1000 is 20. So, 0.02 * 1000 = 20. So, 0.02 * 2000 = 40. Because 20 * 2 = 40. So, yeah, 0.02 * 2000 = 40. Therefore, the daily caloric requirement is 2000 calories.

Alternatively, maybe I can think in terms of proportions. If 2% corresponds to 40 calories, then 1% would correspond to 40 divided by 2, which is 20 calories. Then, 100% would be 20 calories times 100, which is 2000 calories.

Yeah, that method also gives me 2000. So, both ways, I get the same answer.

So, I think that's it. The daily caloric requirement is 2000 calories.

But wait, let me check with another approach. Let's say the total calories are X. So, 2% of X is 40. So, 2/100 * X = 40. So, (2X)/100 = 40. Multiply both sides by 100: 2X = 4000. Divide both sides by 2: X = 2000.

Yep, same answer.

I think that's solid. So, no matter which method I use, I get 2000. So, I think that's the correct answer.

But just to be thorough, let me think if there's any chance that the question is trying to trick me. For example, does it mention anything about the tin being a snack-size, so maybe there's some other consideration? Like, maybe the 40 calories is not the total? But no, the problem says the snack-size tin has 40 calories and that's 2% of the daily requirement. So, the tin's caloric content is 2% of the daily requirement. So, that's straightforward.

No, I don't think there's any trick here. It's a simple percentage problem.

So, conclusion: daily caloric requirement is 2000 calories.

**Final Answer**
The person's daily caloric requirement is \boxed{2000} calories.

To determine a person's daily caloric requirement based on the given information, we begin by understanding that:

> A snack-size tin of peaches contains **40 calories**, and that is **2%** of a person's **daily caloric requirement**.

---

### Step 1: Translate the percentage into a decimal

We know that 2% of the daily requirement is 40 calories. To convert a percentage into its decimal form:

$$
2\% = 0.02
$$

Let the total daily caloric requirement be $ X $. Then, the statement becomes a simple equation:

$$
0.02 \times X = 40
$$

---

### Step 2: Solve the equation

We solve for $ X $ by dividing both sides of the equation by 0.02:

$$
X = \frac{40}{0.02}
$$

$$
X = 2000
$$

---

### Step 3: Verification (Optional, for thoroughness)

Let’s verify by checking that 2% of 2000 equals 40:

$$
0.02 \times 2000 = 40
$$

This confirms the calculation is correct.

---

### Conclusion

Based on the calculation and verification, the person's daily caloric requirement is:

$$
\boxed{2000}
$$

Solution: If 40 calories is equal to $2\%=\frac{2}{100}=\frac{1}{50}$ of a person's daily requirement, then a person's daily caloric requirement is: $$40\cdot 50=\boxed{2000}$$


Explanation:
Okay, so I need to compute 99 squared plus 99 plus 1. Let me think about how to do this mentally. Hmm, 99 is close to 100, which is a nice round number. Maybe I can use some algebraic identities or break it down into parts that are easier to handle. Let me recall that (a - b)^2 is a^2 - 2ab + b^2. But here I have 99 squared, so maybe I can write 99 as 100 - 1. That might be helpful. Let me try that.

So, 99 squared is (100 - 1)^2. Let me expand that. Using the formula (a - b)^2 = a² - 2ab + b², so substituting a = 100 and b = 1, that would be 100² - 2*100*1 + 1². Calculating each term:

100 squared is 10,000. Then 2*100*1 is 200. And 1 squared is 1. So putting it all together: 10,000 - 200 + 1. Let me compute that. 10,000 minus 200 is 9,800, and then adding 1 gives 9,801. So 99 squared is 9,801. Okay, that seems right.

Now, the original problem is 99² + 99 + 1. So I have already calculated 99² as 9,801. Now I need to add 99 and then 1. Let me do that step by step. So adding 99 to 9,801. Let me think. 9,801 + 99. Hmm, 9,801 + 99 is the same as 9,801 + 100 - 1. Because 99 is 100 minus 1. So that would be 9,801 + 100 = 9,901, then subtract 1 gives 9,900. Then I need to add the remaining 1. Wait, the original problem is 99² + 99 + 1. So after adding 99 to 99², which I just did as 9,900, then adding 1 would make it 9,900 + 1 = 9,901. Wait, is that correct?

Let me check again. Let me do 99² + 99 + 1. If I compute each term separately:

99² is 9,801. Then adding 99 gives 9,801 + 99. Let me compute that directly. 9,801 + 99. Breaking it down: 9,801 + 90 = 9,891, then +9 = 9,900. Then adding 1 gives 9,901. So that's 9,901. Wait, but is there a smarter way to do this?

Alternatively, maybe I can factor something. Let me see. The expression is 99² + 99 + 1. Is there a way to factor this? Well, 99² + 99 is 99*(99 + 1) = 99*100. Then adding 1 gives 99*100 + 1. Which is 9,900 + 1 = 9,901. Oh, that's a better way. So instead of expanding 99² first, I can factor out the 99 from the first two terms. That seems easier. Let me confirm:

Original expression: 99² + 99 + 1. Let's group the first two terms: 99² + 99 = 99*(99 + 1) = 99*100. Then adding 1 gives 99*100 + 1. Which is 9,900 + 1 = 9,901. So that works. So that's a more straightforward way. So maybe I should have thought of that first. But either way, both methods lead to the same answer.

Alternatively, could I think of the entire expression as part of a formula? For example, maybe something like n² + n + 1 where n is 99? But I don't recall a specific formula for that. But maybe if I think of it as n² + n + 1 = (n³ - 1)/(n - 1) when n ≠ 1, but that's for geometric series. Wait, let me check:

The formula for the sum of a geometric series is (r^{n} - 1)/(r - 1). If I take r = n and sum from k=0 to 2, it would be 1 + n + n². So yes, that's equal to (n³ - 1)/(n - 1). But in this case, n is 99. So substituting n = 99, we have 1 + 99 + 99² = (99³ - 1)/(99 - 1) = (99³ - 1)/98. But that seems more complicated than necessary. Maybe not helpful for mental calculation. Let me check if that gives the same result. Let's compute (99³ - 1)/98. Well, 99³ is 970,299. Wait, 99*99=9801, then 9801*99. Let me compute 9801*100 - 9801 = 980,100 - 9,801 = 970,299. Then subtract 1 gives 970,298. Divided by 98. Let me compute 970,298 / 98. Let's divide numerator and denominator by 2: 485,149 / 49. Hmm, not sure if that helps. Maybe I can divide 970,298 by 98. Let me try. 98*9,900 = 98*(10,000 - 100) = 980,000 - 9,800 = 970,200. Then 970,298 - 970,200 = 98. So 98*9,900 + 98 = 98*(9,900 + 1) = 98*9,901. Therefore, (99³ - 1)/98 = 9,901. Which matches the previous answer. But that seems more complicated than necessary for mental math. So maybe not the best approach here. So better to stick with the previous methods.

So, going back, the two methods I have are:

Method 1: 99² = 9,801. Then add 99: 9,801 + 99 = 9,900. Then add 1: 9,901.

Method 2: 99² + 99 = 99*100 = 9,900. Then add 1: 9,901.

Either way, same result. Let me check if there's another way. For instance, thinking of 99 as 100 - 1, so maybe expanding the entire expression.

Original expression: 99² + 99 + 1.

Let me write all terms in terms of 100 - 1.

First, 99² is (100 - 1)^2 = 100² - 2*100*1 + 1² = 10,000 - 200 + 1 = 9,801.

Then 99 is 100 - 1. So adding 99 is adding 100 - 1.

So total expression becomes:

9,801 + (100 - 1) + 1 = 9,801 + 100 - 1 + 1 = 9,801 + 100 + 0 = 9,901. So same answer. Therefore, regardless of the method, it's 9,901.

Alternatively, maybe think of the entire expression as (99² + 99 + 1). Let me see if that's a known formula. For example, in some contexts, but I don't recall. But maybe if I think of it as related to a cubic equation? Not sure.

Alternatively, think of 99² + 99 + 1 = 99*(99 + 1) + 1 = 99*100 + 1 = 9,900 + 1 = 9,901. Which is the same as method 2. So, yes, that's straightforward.

Alternatively, maybe think of 99² + 99 + 1 = (99 + 0.5)^2 + something? Not sure. Maybe complicating.

Alternatively, think of 99^2 + 99 + 1 = 99^2 + 99 + 1. If I didn't remember the methods above, maybe do it step by step:

First, compute 99 squared. 99*99. Well, 100*99 = 9,900. Then subtract 1*99, so 9,900 - 99 = 9,801. That's the same as before. Then add 99 to that: 9,801 + 99. 9,801 + 99 is 9,900. Then add 1: 9,901. Yep.

Alternatively, if I didn't remember that 99*99 is 9,801, maybe compute 99*99 as (90 + 9)*(90 + 9). But that might be more complicated. Alternatively, use the formula (a + b)^2 = a² + 2ab + b². But that would be if I consider 99 as 100 - 1, which we already did. So anyway.

Alternatively, if I didn't use any formulas, maybe just do the multiplication:

Compute 99*99:

First, 99*99. Let me do it step by step:

Multiply 99 by 99:

First, compute 99*100 = 9,900. Then subtract 99, so 9,900 - 99 = 9,801. Yep. So that works.

Alternatively, using the standard multiplication:

   99
x 99
-----
   891  (99*9)
+8910   (99*90, shifted one position)
-------
 9801

Which is 9,801. So that's correct.

So, once I have 9,801, adding 99 and 1 is straightforward. So 9,801 + 99 is 9,900. Then adding 1 gives 9,901.

Alternatively, perhaps I can think of 99^2 + 99 + 1 as part of a pattern. For example, if n = 99, then n² + n + 1. If I didn't have a specific formula, but maybe if I think of n³ - 1 = (n - 1)(n² + n + 1). So, n² + n + 1 = (n³ - 1)/(n - 1). As before. So for n = 99, that would be (99³ - 1)/98. But as calculated earlier, that equals 9,901. But that requires knowing that formula and being able to compute 99³ - 1 divided by 98, which seems more complicated, but maybe if I do it step by step:

First, compute 99³. As before, 99³ = 99*99*99. Which is 9,801*99. Let me compute that. 9,801*100 = 980,100. Subtract 9,801: 980,100 - 9,801 = 970,299. Then subtract 1: 970,299 - 1 = 970,298. Then divide by 98: 970,298 / 98. Let me divide 970,298 by 98.

First, note that 98*10,000 = 980,000. But 970,298 is less than that. Let me compute how many times 98 goes into 970,298. Let me divide numerator and denominator by 2: 485,149 / 49. Let me see if 49 goes into 485,149. Well, 49*9,900 = 49*(10,000 - 100) = 490,000 - 4,900 = 485,100. Then 485,149 - 485,100 = 49. Therefore, 49*9,900 + 49 = 49*(9,900 + 1) = 49*9,901. Therefore, 485,149 / 49 = 9,901. Therefore, 970,298 / 98 = 9,901. So yes, same answer. But that seems like a longer process, so not really helpful for mental math unless you remember that formula. So for someone doing this in their head, maybe the first methods are better.

Alternatively, maybe think of 99² + 99 + 1 as (99 + 1)^2 - 99. Wait, let's check:

(99 + 1)^2 = 100² = 10,000. Then subtract 99 gives 10,000 - 99 = 9,901. Oh! That's another way. So that's clever. Let me verify:

Original expression: 99² + 99 + 1.

If I write it as (99 + 1)^2 - 99. Let's expand (99 + 1)^2 = 99² + 2*99*1 + 1² = 99² + 198 + 1. So then subtracting 99 would be 99² + 198 + 1 - 99 = 99² + 99 + 1, which matches the original expression. Therefore, yes, 99² + 99 + 1 equals (99 + 1)^2 - 99 = 100² - 99 = 10,000 - 99 = 9,901. That seems like an elegant way to think about it. So that's another method.

So this method is perhaps even simpler. Because 100 squared is easy, 10,000. Then subtract 99. Which is 10,000 - 99 = 9,901. So that's straightforward. Therefore, that's a very clever way.

So, in summary, multiple approaches lead to the same answer. The key was recognizing that 99 is one less than 100, and using that to simplify calculations. Whether by expanding squares, factoring, or manipulating the expression into a form that uses 100, the answer is consistently 9,901.

Let me just verify once again with another method. Let me add all the numbers step by step, without any algebra.

First, 99 squared is 9,801. Then adding 99: 9,801 + 99. Let's add 9,801 + 99. Since 9,801 + 99 = 9,801 + 100 - 1 = 9,901 - 1 = 9,900. Wait, no, 9,801 + 100 is 9,901, then subtract 1 gives 9,900. Then add the remaining 1? Wait, original problem is 99² + 99 + 1, so after adding 99 to 99², which gives 9,900, then adding 1 gives 9,901. So yes. Correct.

Alternatively, if I didn't break it down, but just added them all together:

9,801 (from 99²) + 99 (from the second term) + 1 (third term) = 9,801 + 99 + 1. Let me add 99 and 1 first: 99 + 1 = 100. Then add that to 9,801: 9,801 + 100 = 9,901. Oh! That's even more straightforward. So, 99² + 99 + 1 = 9,801 + (99 + 1) = 9,801 + 100 = 9,901. That's a very simple way. So that's another way to look at it.

Hence, regardless of the method, the answer is 9,901. So I can be confident that this is the correct result.

Just to make sure I haven't made any arithmetic errors, let me do a quick check with smaller numbers. Let's say instead of 99, use 2. Compute 2² + 2 + 1 = 4 + 2 + 1 = 7. Using the method above: (2 + 1)^2 - 2 = 9 - 2 = 7. Correct. Another example: 3² + 3 + 1 = 9 + 3 + 1 = 13. Using the formula: (3 + 1)^2 - 3 = 16 - 3 = 13. Correct. So the formula works. Therefore, applying it to 99 gives 100² - 99 = 10,000 - 99 = 9,901. So that's solid.

Therefore, after verifying with multiple methods and checking with smaller numbers, I can confirm that the answer is indeed 9,901.

**Final Answer**
The value of $99^2 + 99 + 1$ is \boxed{9901}.

To compute $99^2 + 99 + 1$ mentally, we can leverage the fact that $99$ is close to $100$, which makes the calculations simpler.

---

### Step 1: Recognize a Pattern

Observe that:
$$
99^2 + 99 + 1 = (99 + 1)^2 - 99
$$

This identity is useful because:
$$
(99 + 1)^2 = 100^2 = 10,000
$$
Then subtracting 99 gives:
$$
10,000 - 99 = 9,901
$$

So,
$$
99^2 + 99 + 1 = \boxed{9901}
$$

---

### Step 2: Verify with Another Method

We can check the result by computing directly.

- First, calculate $99^2$:
  $$
  99^2 = (100 - 1)^2 = 100^2 - 2 \cdot 100 + 1 = 10,000 - 200 + 1 = 9,801
  $$

- Then add 99 and 1:
  $$
  9,801 + 99 + 1 = 9,801 + 100 = 9,901
  $$

---

### Final Answer

$$
\boxed{9901}
$$

[Assistant]
Explanation:
Okay, so I need to compute 99 squared plus 99 plus 1. Let me think about how to do this mentally. Hmm, 99 is close to 100, which is a nice round number. Maybe I can use some algebraic identities or break it down into parts that are easier to handle. Let me recall that (a - b)^2 is a^2 - 2ab + b^2. But here I have 99 squared, so maybe I can write 99 as 100 - 1. That might be helpful. Let me try that.

So, 99 squared is (100 - 1)^2. Let me expand that. Using the formula (a - b)^2 = a² - 2ab + b², so substituting a = 100 and b = 1, that would be 100² - 2*100*1 + 1². Calculating each term:

100 squared is 10,000. Then 2*100*1 is 200. And 1 squared is 1. So putting it all together: 10,000 - 200 + 1. Let me compute that. 10,000 minus 200 is 9,800, and then adding 1 gives 9,801. So 99 squared is 9,801. Okay, that seems right.

Now, the original problem is 99² + 99 + 1. So I have already calculated 99² as 9,801. Now I need to add 99 and then 1. Let me do that step by step. So adding 99 to 9,801. Let me think. 9,801 + 99. Hmm, 9,801 + 99 is the same as 9,801 + 100 - 1. Because 99 is 100 minus 1. So that would be 9,801 + 100 = 9,901, then subtract 1 gives 9,900. Then I need to add the remaining 1. Wait, the original problem is 99² + 99 + 1. So after adding 99 to 99², which I just did as 9,900, then adding 1 would make it 9,900 + 1 = 9,901. Wait, is that correct?

Let me check again. Let me do 99² + 99 + 1. If I compute each term separately:

99² is 9,801. Then adding 99 gives 9,801 + 99. Let me compute that directly. 9,801 + 99. Breaking it down: 9,801 + 90 = 9,891, then +9 = 9,900. Then adding 1 gives 9,901. So that's 9,901. Wait, but is there a smarter way to do this?

Alternatively, maybe I can factor something. Let me see. The expression is 99² + 99 + 1. Is there a way to factor this? Well, 99² + 99 is 99*(99 + 1) = 99*100. Then adding 1 gives 99*100 + 1. Which is 9,900 + 1 = 9,901. Oh, that's a better way. So instead of expanding 99² first, I can factor out the 99 from the first two terms. That seems easier. Let me confirm:

Original expression: 99² + 99 + 1. Let's group the first two terms: 99² + 99 = 99*(99 + 1) = 99*100. Then adding 1 gives 99*100 + 1. Which is 9,900 + 1 = 9,901. So that works. So that's a more straightforward way. So maybe I should have thought of that first. But either way, both methods lead to the same answer.

Alternatively, could I think of the entire expression as part of a formula? For example, maybe something like n² + n + 1 where n is 99? But I don't recall a specific formula for that. But maybe if I think of it as n² + n + 1 = (n³ - 1)/(n - 1) when n ≠ 1, but that's for geometric series. Wait, let me check:

The formula for the sum of a geometric series is (r^{n} - 1)/(r - 1). If I take r = n and sum from k=0 to 2, it would be 1 + n + n². So yes, that's equal to (n³ - 1)/(n - 1). But in this case, n is 99. So substituting n = 99, we have 1 + 99 + 99² = (99³ - 1)/(99 - 1) = (99³ - 1)/98. But that seems more complicated than necessary. Maybe not helpful for mental calculation. Let me check if that gives the same result. Let's compute (99³ - 1)/98. Well, 99³ is 970,299. Wait, 99*99=9801, then 9801*99. Let me compute 9801*100 - 9801 = 980,100 - 9,801 = 970,299. Then subtract 1 gives 970,298. Divided by 98. Let me compute 970,298 / 98. Let's divide numerator and denominator by 2: 485,149 / 49. Hmm, not sure if that helps. Maybe I can divide 970,298 by 98. Let me try. 98*9,900 = 98*(10,000 - 100) = 980,000 - 9,800 = 970,200. Then 970,298 - 970,200 = 98. So 98*9,900 + 98 = 98*(9,900 + 1) = 98*9,901. Therefore, (99³ - 1)/98 = 9,901. Which matches the previous answer. But that seems more complicated than necessary for mental math. So maybe not the best approach here. So better to stick with the previous methods.

So, going back, the two methods I have are:

Method 1: 99² = 9,801. Then add 99: 9,801 + 99 = 9,900. Then add 1: 9,901.

Method 2: 99² + 99 = 99*100 = 9,900. Then add 1: 9,901.

Either way, same result. Let me check if there's another way. For instance, thinking of 99 as 100 - 1, so maybe expanding the entire expression.

Original expression: 99² + 99 + 1.

Let me write all terms in terms of 100 - 1.

First, 99² is (100 - 1)^2 = 100² - 2*100*1 + 1² = 10,000 - 200 + 1 = 9,801.

Then 99 is 100 - 1. So adding 99 is adding 100 - 1.

So total expression becomes:

9,801 + (100 - 1) + 1 = 9,801 + 100 - 1 + 1 = 9,801 + 100 + 0 = 9,901. So same answer. Therefore, regardless of the method, it's 9,901.

Alternatively, maybe think of the entire expression as (99² + 99 + 1). Let me see if that's a known formula. For example, in some contexts, but I don't recall. But maybe if I think of it as related to a cubic equation? Not sure.

Alternatively, think of 99² + 99 + 1 = 99*(99 + 1) + 1 = 99*100 + 1 = 9,900 + 1 = 9,901. Which is the same as method 2. So, yes, that's straightforward.

Alternatively, maybe think of 99² + 99 + 1 = (99 + 0.5)^2 + something? Not sure. Maybe complicating.

Alternatively, think of 99^2 + 99 + 1 = 99^2 + 99 + 1. If I didn't remember the methods above, maybe do it step by step:

First, compute 99 squared. 99*99. Well, 100*99 = 9,900. Then subtract 1*99, so 9,900 - 99 = 9,801. That's the same as before. Then add 99 to that: 9,801 + 99. 9,801 + 99 is 9,900. Then add 1: 9,901. Yep.

Alternatively, if I didn't remember that 99*99 is 9,801, maybe compute 99*99 as (90 + 9)*(90 + 9). But that might be more complicated. Alternatively, use the formula (a + b)^2 = a² + 2ab + b². But that would be if I consider 99 as 100 - 1, which we already did. So anyway.

Alternatively, if I didn't use any formulas, maybe just do the multiplication:

Compute 99*99:

First, 99*99. Let me do it step by step:

Multiply 99 by 99:

First, compute 99*100 = 9,900. Then subtract 99, so 9,900 - 99 = 9,801. Yep. So that works.

Alternatively, using the standard multiplication:

   99
x 99
-----
   891  (99*9)
+8910   (99*90, shifted one position)
-------
 9801

Which is 9,801. So that's correct.

So, once I have 9,801, adding 99 and 1 is straightforward. So 9,801 + 99 is 9,900. Then adding 1 gives 9,901.

Alternatively, perhaps I can think of 99^2 + 99 + 1 as part of a pattern. For example, if n = 99, then n² + n + 1. If I didn't have a specific formula, but maybe if I think of n³ - 1 = (n - 1)(n² + n + 1). So, n² + n + 1 = (n³ - 1)/(n - 1). As before. So for n = 99, that would be (99³ - 1)/98. But as calculated earlier, that equals 9,901. But that requires knowing that formula and being able to compute 99³ - 1 divided by 98, which seems more complicated, but maybe if I do it step by step:

First, compute 99³. As before, 99³ = 99*99*99. Which is 9,801*99. Let me compute that. 9,801*100 = 980,100. Subtract 9,801: 980,100 - 9,801 = 970,299. Then subtract 1: 970,299 - 1 = 970,298. Then divide by 98: 970,298 / 98. Let me divide 970,298 by 98.

First, note that 98*10,000 = 980,000. But 970,298 is less than that. Let me compute how many times 98 goes into 970,298. Let me divide numerator and denominator by 2: 485,149 / 49. Let me see if 49 goes into 485,149. Well, 49*9,900 = 49*(10,000 - 100) = 490,000 - 4,900 = 485,100. Then 485,149 - 485,100 = 49. Therefore, 49*9,900 + 49 = 49*(9,900 + 1) = 49*9,901. Therefore, 485,149 / 49 = 9,901. Therefore, 970,298 / 98 = 9,901. So yes, same answer. But that seems like a longer process, so not really helpful for mental math unless you remember that formula. So for someone doing this in their head, maybe the first methods are better.

Alternatively, maybe think of 99² + 99 + 1 as (99 + 1)^2 - 99. Wait, let's check:

(99 + 1)^2 = 100² = 10,000. Then subtract 99 gives 10,000 - 99 = 9,901. Oh! That's another way. So that's clever. Let me verify:

Original expression: 99² + 99 + 1.

If I write it as (99 + 1)^2 - 99. Let's expand (99 + 1)^2 = 99² + 2*99*1 + 1² = 99² + 198 + 1. So then subtracting 99 would be 99² + 198 + 1 - 99 = 99² + 99 + 1, which matches the original expression. Therefore, yes, 99² + 99 + 1 equals (99 + 1)^2 - 99 = 100² - 99 = 10,000 - 99 = 9,901. That seems like an elegant way to think about it. So that's another method.

So this method is perhaps even simpler. Because 100 squared is easy, 10,000. Then subtract 99. Which is 10,000 - 99 = 9,901. So that's straightforward. Therefore, that's a very clever way.

So, in summary, multiple approaches lead to the same answer. The key was recognizing that 99 is one less than 100, and using that to simplify calculations. Whether by expanding squares, factoring, or manipulating the expression into a form that uses 100, the answer is consistently 9,901.

Let me just verify once again with another method. Let me add all the numbers step by step, without any algebra.

First, 99 squared is 9,801. Then adding 99: 9,801 + 99. Let's add 9,801 + 99. Since 9,801 + 99 = 9,801 + 100 - 1 = 9,901 - 1 = 9,900. Wait, no, 9,801 + 100 is 9,901, then subtract 1 gives 9,900. Then add the remaining 1? Wait, original problem is 99² + 99 + 1, so after adding 99 to 99², which gives 9,900, then adding 1 gives 9,901. So yes. Correct.

Alternatively, if I didn't break it down, but just added them all together:

9,801 (from 99²) + 99 (from the second term) + 1 (third term) = 9,801 + 99 + 1. Let me add 99 and 1 first: 99 + 1 = 100. Then add that to 9,801: 9,801 + 100 = 9,901. Oh! That's even more straightforward. So, 99² + 99 + 1 = 9,801 + (99 + 1) = 9,801 + 100 = 9,901. That's a very simple way. So that's another way to look at it.

Hence, regardless of the method, the answer is 9,901. So I can be confident that this is the correct result.

Just to make sure I haven't made any arithmetic errors, let me do a quick check with smaller numbers. Let's say instead of 99, use 2. Compute 2² + 2 + 1 = 4 + 2 + 1 = 7. Using the method above: (2 + 1)^2 - 2 = 9 - 2 = 7. Correct. Another example: 3² + 3 + 1 = 9 + 3 + 1 = 13. Using the formula: (3 + 1)^2 - 3 = 16 - 3 = 13. Correct. So the formula works. Therefore, applying it to 99 gives 100² - 99 = 10,000 - 99 = 9,901. So that's solid.

Therefore, after verifying with multiple methods and checking with smaller numbers, I can confirm that the answer is indeed 9,901.

**Final Answer**
The value of $99^2 + 99 + 1$ is \boxed{9901}.

To compute $99^2 + 99 + 1$ mentally, we can leverage the fact that $99$ is close to $100$, which makes the calculations simpler.

---

### Step 1: Recognize a Pattern

Observe that:
$$
99^2 + 99 + 1 = (99 + 1)^2 - 99
$$

This identity is useful because:
$$
(99 + 1)^2 = 100^2 = 10,000
$$
Then subtracting 99 gives:
$$
10,000 - 99 = 9,901
$$

So,
$$
99^2 + 99 + 1 = \boxed{9901}
$$

---

### Step 2: Verify with Another Method

We can check the result by computing directly.

- First, calculate $99^2$:
  $$
  99^2 = (100 - 1)^2 = 100^2 - 2 \cdot 100 + 1 = 10,000 - 200 + 1 = 9,801
  $$

- Then add 99 and 1:
  $$
  9,801 + 99 + 1 = 9,801 + 100 = 9,901
  $$

---

### Final Answer

$$
\boxed{9901}
$$

Solution:
Factoring the first two terms, we have:

$99^2+99+1=99(99+1)+1=99\cdot 100+1=9900+1=\boxed{9901}$.


Explanation:
Okay, so I need to figure out what 1 raised to the power of (2^235423523) is. Let me think about this step by step.

First, I know that any number raised to a power is just that number multiplied by itself that many times. But when the base is 1, no matter how many times you multiply 1 by itself, the result should still be 1, right? Like 1^2 is 1*1=1, 1^3 is 1*1*1=1, and so on. So, even if the exponent is a huge number, like 2^235423523, which is an astronomically large number, does that still hold?

Wait, let me make sure. Is there any case where 1 raised to some power isn't 1? Hmm... Well, 1 to any real exponent is still 1. Because multiplying 1 by itself any number of times doesn't change its value. Even if the exponent is a fraction or a decimal, like 1^(1/2) is still 1, since the square root of 1 is 1. Similarly, 1^(-2) is 1/(1^2) = 1/1 = 1. So regardless of the exponent, positive, negative, integer, fractional, 1 to that power is 1.

But hold on, the exponent here is 2^235423523. Let me check if that exponent is a valid exponent. Well, exponents can be any real number, but in this case, it's an integer. 2 raised to any positive integer power is just a positive integer. So, even though the exponent is extremely large, it's still a valid integer exponent.

Therefore, 1 raised to any integer power, no matter how big, should still be 1. So, 1^(2^235423523) is just 1.

Wait, but maybe there's some trick here? Like, maybe the exponent is zero? But no, 2^anything is definitely not zero. 2^anything is at least 2^0 = 1. So the exponent is a positive integer.

Alternatively, could there be some context where 1^something is not 1? For example, in modular arithmetic? But the problem doesn't specify any modulus. So unless there's an implicit modulus, which there isn't, the answer should just be 1.

Let me think again. If I have 1^n where n is any positive integer, the result is 1. If n is zero, 1^0 is 1 as well. If n is negative, 1^n is still 1. So regardless of n, 1^n is 1.

Therefore, even if n is 2^235423523, which is a massive number, the answer is 1.

I think that's it. The key insight is recognizing that 1 to any power is always 1. The exponent here is just an extremely large number, but the base is 1, so the result remains 1. There's no need to compute the exponent or do anything else.

But let me verify with some examples. Let's take smaller exponents first.

1^1 = 1.

1^2 = 1*1 = 1.

1^10 = 1.

Even 1^1000000 is 1. So, regardless of how big the exponent is, as long as the base is 1, the result is 1.

Therefore, even with an exponent of 2^235423523, which is 2 raised to a very large exponent, the answer is still 1.

So, I think the answer is 1.

**Final Answer**
\boxed{1}

To evaluate the expression $ 1^{(2^{235423523})} $, we examine the behavior of the number 1 under exponentiation.

---

### Understanding the Base and Exponent

The base of the expression is $ 1 $, and the exponent is $ 2^{235423523} $, which is a **very large positive integer**. The exponent's size, although formidable, is irrelevant in this context.

---

### Key Property of Exponentiation with Base 1

A fundamental mathematical fact is that for **any real number $ a $** and **any exponent $ n $**, the following property holds:

$$
1^n = 1
$$

This is because $ 1 $ multiplied by itself any number of times remains $ 1 $. Regardless of the magnitude of the exponent, this rule applies. Specifically, even if the exponent is an astronomically large number like $ 2^{235423523} $, the result of $ 1^{(2^{235423523})} $ is still just $ 1 $.

---

### Additional Considerations

- **No Modulus or Contextual Constraints**: Since the problem does not introduce a modulus or alternate arithmetic system (like modular arithmetic), the result is straightforward.
- **Exponent Validity**: The exponent $ 2^{235423523} $ is a well-defined integer (positive), ensuring the expression is valid in standard arithmetic.

---

### Final Conclusion

Because the base is 1 and the exponent is a positive integer, the expression simplifies directly to:

$$
\boxed{1}
$$

[Assistant]
Explanation:
Okay, so I need to figure out what 1 raised to the power of (2^235423523) is. Let me think about this step by step.

First, I know that any number raised to a power is just that number multiplied by itself that many times. But when the base is 1, no matter how many times you multiply 1 by itself, the result should still be 1, right? Like 1^2 is 1*1=1, 1^3 is 1*1*1=1, and so on. So, even if the exponent is a huge number, like 2^235423523, which is an astronomically large number, does that still hold?

Wait, let me make sure. Is there any case where 1 raised to some power isn't 1? Hmm... Well, 1 to any real exponent is still 1. Because multiplying 1 by itself any number of times doesn't change its value. Even if the exponent is a fraction or a decimal, like 1^(1/2) is still 1, since the square root of 1 is 1. Similarly, 1^(-2) is 1/(1^2) = 1/1 = 1. So regardless of the exponent, positive, negative, integer, fractional, 1 to that power is 1.

But hold on, the exponent here is 2^235423523. Let me check if that exponent is a valid exponent. Well, exponents can be any real number, but in this case, it's an integer. 2 raised to any positive integer power is just a positive integer. So, even though the exponent is extremely large, it's still a valid integer exponent.

Therefore, 1 raised to any integer power, no matter how big, should still be 1. So, 1^(2^235423523) is just 1.

Wait, but maybe there's some trick here? Like, maybe the exponent is zero? But no, 2^anything is definitely not zero. 2^anything is at least 2^0 = 1. So the exponent is a positive integer.

Alternatively, could there be some context where 1^something is not 1? For example, in modular arithmetic? But the problem doesn't specify any modulus. So unless there's an implicit modulus, which there isn't, the answer should just be 1.

Let me think again. If I have 1^n where n is any positive integer, the result is 1. If n is zero, 1^0 is 1 as well. If n is negative, 1^n is still 1. So regardless of n, 1^n is 1.

Therefore, even if n is 2^235423523, which is a massive number, the answer is 1.

I think that's it. The key insight is recognizing that 1 to any power is always 1. The exponent here is just an extremely large number, but the base is 1, so the result remains 1. There's no need to compute the exponent or do anything else.

But let me verify with some examples. Let's take smaller exponents first.

1^1 = 1.

1^2 = 1*1 = 1.

1^10 = 1.

Even 1^1000000 is 1. So, regardless of how big the exponent is, as long as the base is 1, the result is 1.

Therefore, even with an exponent of 2^235423523, which is 2 raised to a very large exponent, the answer is still 1.

So, I think the answer is 1.

**Final Answer**
\boxed{1}

To evaluate the expression $ 1^{(2^{235423523})} $, we examine the behavior of the number 1 under exponentiation.

---

### Understanding the Base and Exponent

The base of the expression is $ 1 $, and the exponent is $ 2^{235423523} $, which is a **very large positive integer**. The exponent's size, although formidable, is irrelevant in this context.

---

### Key Property of Exponentiation with Base 1

A fundamental mathematical fact is that for **any real number $ a $** and **any exponent $ n $**, the following property holds:

$$
1^n = 1
$$

This is because $ 1 $ multiplied by itself any number of times remains $ 1 $. Regardless of the magnitude of the exponent, this rule applies. Specifically, even if the exponent is an astronomically large number like $ 2^{235423523} $, the result of $ 1^{(2^{235423523})} $ is still just $ 1 $.

---

### Additional Considerations

- **No Modulus or Contextual Constraints**: Since the problem does not introduce a modulus or alternate arithmetic system (like modular arithmetic), the result is straightforward.
- **Exponent Validity**: The exponent $ 2^{235423523} $ is a well-defined integer (positive), ensuring the expression is valid in standard arithmetic.

---

### Final Conclusion

Because the base is 1 and the exponent is a positive integer, the expression simplifies directly to:

$$
\boxed{1}
$$

Solution: One raised to any power is one, so our answer is $\boxed{1}.$


Explanation:
Okay, so I need to solve the equation $2^8 = 4^x$ and find the value of $x$. Let me think about how to approach this.

First, I remember that when dealing with exponents, it's helpful to express both sides of the equation with the same base. That way, I can set the exponents equal to each other. Let me check if that's possible here.

The left side of the equation is $2^8$. The right side is $4^x$. I know that 4 is a power of 2. Specifically, 4 is $2^2$. Maybe I can rewrite $4^x$ in terms of base 2. Let me try that.

So, substituting $4$ with $2^2$, the equation becomes:
$$
2^8 = (2^2)^x
$$
Now, I recall the exponent rule that says $(a^b)^c = a^{b \cdot c}$. Applying that to the right side, I can simplify $(2^2)^x$ to $2^{2x}$. So now the equation is:
$$
2^8 = 2^{2x}
$$
Since the bases are the same and both sides are equal, their exponents must be equal as well. Therefore, I can set the exponents equal to each other:
$$
8 = 2x
$$
To solve for $x$, I just need to divide both sides of the equation by 2:
$$
x = \frac{8}{2} = 4
$$
Wait, let me double-check my steps to make sure I didn't make any mistakes.

Starting with the original equation: $2^8 = 4^x$. Converted 4 to $2^2$, so $4^x$ becomes $(2^2)^x = 2^{2x}$. Then equate the exponents because the bases are equal: $8 = 2x$. Solving for $x$ gives $x = 4$.

Let me verify by plugging $x = 4$ back into the original equation.

Left side: $2^8 = 256$.

Right side: $4^4 = (2^2)^4 = 2^{8} = 256$.

Yes, both sides equal 256, so the solution $x = 4$ is correct.

Alternatively, maybe there's another way to approach this problem? Let me think.

Another method could be taking logarithms of both sides. Let me try that as a check.

Taking natural logarithm on both sides:
$$
\ln(2^8) = \ln(4^x)
$$
Using the logarithm power rule $\ln(a^b) = b\ln(a)$, this becomes:
$$
8\ln(2) = x\ln(4)
$$
Then solving for $x$ gives:
$$
x = \frac{8\ln(2)}{\ln(4)}
$$
But since $\ln(4)$ is $\ln(2^2) = 2\ln(2)$, substitute that in:
$$
x = \frac{8\ln(2)}{2\ln(2)} = \frac{8}{2} = 4
$$
Same result. So that confirms the answer again.

Alternatively, maybe using common logarithms? Let me try that just to be thorough.

Taking log base 10:
$$
\log(2^8) = \log(4^x)
$$
Which would be:
$$
8\log(2) = x\log(4)
$$
And similarly, $\log(4)$ is $\log(2^2) = 2\log(2)$, so:
$$
x = \frac{8\log(2)}{2\log(2)} = 4
$$
Again, same answer. So regardless of the logarithm base, I end up with $x = 4$.

Is there a way to see this without using logarithms? Maybe by recognizing the relationship between the exponents?

Since 4 is 2 squared, then 4^x is (2^2)^x = 2^{2x}. Therefore, 2^8 = 2^{2x} implies 8 = 2x, so x = 4.

That's essentially what I did first. So all the methods lead to the same conclusion.

Another way to think about it: Let me express both sides in terms of powers of 2.

Original equation: $2^8 = 4^x$.

Since 4 is 2 squared, 4^x is (2^2)^x = 2^{2x}. Therefore, 2^8 equals 2^{2x}.

So, the exponents must be equal: 8 = 2x. Solving for x gives x = 4.

Yes, that's straightforward.

Alternatively, maybe if I didn't remember the exponent rules, I could think in terms of repeated multiplication?

Let me try that.

Left side: 2^8 is 2 multiplied by itself 8 times: 2*2*2*2*2*2*2*2.

Right side: 4^x is 4 multiplied by itself x times. But 4 is 2*2, so 4^x is (2*2)^x.

Which is 2^x * 2^x = 2^{2x}.

Therefore, 2^8 = 2^{2x}, so exponents equal: 8 = 2x.

Same result.

Alternatively, maybe plugging in numbers for x?

If I try x = 4, then 4^4 = 256, which is 2^8.

If I try x = 3, 4^3 = 64, which is 2^6.

x = 5, 4^5 = 1024, which is 2^10.

So, x=4 gives exactly 2^8.

Therefore, the answer is 4.

I think that's thorough enough. All methods lead to x=4.

Just to recap:

1. Express both sides with the same base.

2. Set exponents equal.

3. Solve for x.

4. Verify by substitution.

All steps confirm that x=4 is the correct answer.

I don't see any flaws in the reasoning. Therefore, I'm confident that x is 4.

**Final Answer**
The value of $x$ is \boxed{4}.

To solve the equation:

$$
2^8 = 4^x
$$

we begin by noting that both sides involve exponential expressions. To simplify and solve, we aim to express both sides with the same base. Since $4 = 2^2$, we can rewrite the right-hand side using base 2:

$$
4^x = (2^2)^x
$$

Applying the exponent rule $(a^b)^c = a^{bc}$, we get:

$$
(2^2)^x = 2^{2x}
$$

So the original equation becomes:

$$
2^8 = 2^{2x}
$$

Now that both sides have the same base, we can equate the exponents:

$$
8 = 2x
$$

Solving for $x$:

$$
x = \frac{8}{2} = 4
$$

---

**Verification:**

Let’s check the result by substituting $x = 4$ back into the original equation:

- Left-hand side: $2^8 = 256$
- Right-hand side: $4^4 = (2^2)^4 = 2^{8} = 256$

Both sides match, confirming that the solution is correct.

---

**Alternative Method (Using Logarithms):**

We re-express the original equation using logarithms:

$$
\ln(2^8) = \ln(4^x)
$$

Apply the logarithmic identity $\ln(a^b) = b \ln a$:

$$
8 \ln 2 = x \ln 4
$$

Since $\ln 4 = \ln(2^2) = 2 \ln 2$, we substitute:

$$
8 \ln 2 = x (2 \ln 2)
$$

Dividing both sides by $\ln 2$:

$$
8 = 2x \quad \Rightarrow \quad x = 4
$$

---

**Conclusion:**

Whether through direct comparison of exponents or using logarithms, we find that the value of $x$ that satisfies the equation is:

$$
\boxed{4}
$$

[Assistant]
Explanation:
Okay, so I need to solve the equation $2^8 = 4^x$ and find the value of $x$. Let me think about how to approach this.

First, I remember that when dealing with exponents, it's helpful to express both sides of the equation with the same base. That way, I can set the exponents equal to each other. Let me check if that's possible here.

The left side of the equation is $2^8$. The right side is $4^x$. I know that 4 is a power of 2. Specifically, 4 is $2^2$. Maybe I can rewrite $4^x$ in terms of base 2. Let me try that.

So, substituting $4$ with $2^2$, the equation becomes:
$$
2^8 = (2^2)^x
$$
Now, I recall the exponent rule that says $(a^b)^c = a^{b \cdot c}$. Applying that to the right side, I can simplify $(2^2)^x$ to $2^{2x}$. So now the equation is:
$$
2^8 = 2^{2x}
$$
Since the bases are the same and both sides are equal, their exponents must be equal as well. Therefore, I can set the exponents equal to each other:
$$
8 = 2x
$$
To solve for $x$, I just need to divide both sides of the equation by 2:
$$
x = \frac{8}{2} = 4
$$
Wait, let me double-check my steps to make sure I didn't make any mistakes.

Starting with the original equation: $2^8 = 4^x$. Converted 4 to $2^2$, so $4^x$ becomes $(2^2)^x = 2^{2x}$. Then equate the exponents because the bases are equal: $8 = 2x$. Solving for $x$ gives $x = 4$.

Let me verify by plugging $x = 4$ back into the original equation.

Left side: $2^8 = 256$.

Right side: $4^4 = (2^2)^4 = 2^{8} = 256$.

Yes, both sides equal 256, so the solution $x = 4$ is correct.

Alternatively, maybe there's another way to approach this problem? Let me think.

Another method could be taking logarithms of both sides. Let me try that as a check.

Taking natural logarithm on both sides:
$$
\ln(2^8) = \ln(4^x)
$$
Using the logarithm power rule $\ln(a^b) = b\ln(a)$, this becomes:
$$
8\ln(2) = x\ln(4)
$$
Then solving for $x$ gives:
$$
x = \frac{8\ln(2)}{\ln(4)}
$$
But since $\ln(4)$ is $\ln(2^2) = 2\ln(2)$, substitute that in:
$$
x = \frac{8\ln(2)}{2\ln(2)} = \frac{8}{2} = 4
$$
Same result. So that confirms the answer again.

Alternatively, maybe using common logarithms? Let me try that just to be thorough.

Taking log base 10:
$$
\log(2^8) = \log(4^x)
$$
Which would be:
$$
8\log(2) = x\log(4)
$$
And similarly, $\log(4)$ is $\log(2^2) = 2\log(2)$, so:
$$
x = \frac{8\log(2)}{2\log(2)} = 4
$$
Again, same answer. So regardless of the logarithm base, I end up with $x = 4$.

Is there a way to see this without using logarithms? Maybe by recognizing the relationship between the exponents?

Since 4 is 2 squared, then 4^x is (2^2)^x = 2^{2x}. Therefore, 2^8 = 2^{2x} implies 8 = 2x, so x = 4.

That's essentially what I did first. So all the methods lead to the same conclusion.

Another way to think about it: Let me express both sides in terms of powers of 2.

Original equation: $2^8 = 4^x$.

Since 4 is 2 squared, 4^x is (2^2)^x = 2^{2x}. Therefore, 2^8 equals 2^{2x}.

So, the exponents must be equal: 8 = 2x. Solving for x gives x = 4.

Yes, that's straightforward.

Alternatively, maybe if I didn't remember the exponent rules, I could think in terms of repeated multiplication?

Let me try that.

Left side: 2^8 is 2 multiplied by itself 8 times: 2*2*2*2*2*2*2*2.

Right side: 4^x is 4 multiplied by itself x times. But 4 is 2*2, so 4^x is (2*2)^x.

Which is 2^x * 2^x = 2^{2x}.

Therefore, 2^8 = 2^{2x}, so exponents equal: 8 = 2x.

Same result.

Alternatively, maybe plugging in numbers for x?

If I try x = 4, then 4^4 = 256, which is 2^8.

If I try x = 3, 4^3 = 64, which is 2^6.

x = 5, 4^5 = 1024, which is 2^10.

So, x=4 gives exactly 2^8.

Therefore, the answer is 4.

I think that's thorough enough. All methods lead to x=4.

Just to recap:

1. Express both sides with the same base.

2. Set exponents equal.

3. Solve for x.

4. Verify by substitution.

All steps confirm that x=4 is the correct answer.

I don't see any flaws in the reasoning. Therefore, I'm confident that x is 4.

**Final Answer**
The value of $x$ is \boxed{4}.

To solve the equation:

$$
2^8 = 4^x
$$

we begin by noting that both sides involve exponential expressions. To simplify and solve, we aim to express both sides with the same base. Since $4 = 2^2$, we can rewrite the right-hand side using base 2:

$$
4^x = (2^2)^x
$$

Applying the exponent rule $(a^b)^c = a^{bc}$, we get:

$$
(2^2)^x = 2^{2x}
$$

So the original equation becomes:

$$
2^8 = 2^{2x}
$$

Now that both sides have the same base, we can equate the exponents:

$$
8 = 2x
$$

Solving for $x$:

$$
x = \frac{8}{2} = 4
$$

---

**Verification:**

Let’s check the result by substituting $x = 4$ back into the original equation:

- Left-hand side: $2^8 = 256$
- Right-hand side: $4^4 = (2^2)^4 = 2^{8} = 256$

Both sides match, confirming that the solution is correct.

---

**Alternative Method (Using Logarithms):**

We re-express the original equation using logarithms:

$$
\ln(2^8) = \ln(4^x)
$$

Apply the logarithmic identity $\ln(a^b) = b \ln a$:

$$
8 \ln 2 = x \ln 4
$$

Since $\ln 4 = \ln(2^2) = 2 \ln 2$, we substitute:

$$
8 \ln 2 = x (2 \ln 2)
$$

Dividing both sides by $\ln 2$:

$$
8 = 2x \quad \Rightarrow \quad x = 4
$$

---

**Conclusion:**

Whether through direct comparison of exponents or using logarithms, we find that the value of $x$ that satisfies the equation is:

$$
\boxed{4}
$$

Solution: Rewrite $4$ as $2^2$ to find $4^x=2^{2x}$.  Since $2^8=2^{2x}$, we have $2x=8$ which implies $x=\boxed{4}$.


Explanation:
Okay, so I need to find the value of x that makes the equation 2³ times 3ˣ equal to 72. Let me start by writing down the equation again to make sure I have it right: 2³ · 3ˣ = 72. Alright, let me break this down step by step.

First, I know that 2³ is 8 because 2 times 2 is 4, and 4 times 2 is 8. So substituting that into the equation, it becomes 8 · 3ˣ = 72. Now, I need to solve for x. Hmm, how do I isolate x here? Well, since 3ˣ is being multiplied by 8, I should probably divide both sides of the equation by 8 to get 3ˣ by itself. Let me try that.

Dividing both sides by 8 gives me 3ˣ = 72 / 8. Let me calculate 72 divided by 8. 8 times 9 is 72, so 72 divided by 8 is 9. So now the equation simplifies to 3ˣ = 9. Okay, now I need to solve for x in this equation.

Since the base here is 3 and the right side is 9, I should recall that 9 is a power of 3. Specifically, 3 squared is 9 because 3 times 3 is 9. So that means 3² = 9. Therefore, if 3ˣ equals 9, then x must be 2. Let me check my work to make sure I didn't make a mistake.

Starting with x = 2, plugging back into the original equation: 2³ · 3². 2³ is 8 and 3² is 9. Multiplying those together, 8 times 9 is indeed 72, which matches the right side of the equation. So that checks out.

Wait, but let me think again. Is there another way to approach this problem? Maybe using logarithms? Let me try that method to confirm.

Starting again with the equation 3ˣ = 9. Taking the logarithm of both sides. Since the base is 3, maybe using logarithm base 3? The logarithm base 3 of 3ˣ is x, and the logarithm base 3 of 9 is log₃(9). Since 9 is 3², log base 3 of 3² is 2. Therefore, x = 2. Same result.

Alternatively, if I use natural logarithm or common logarithm, I can apply the change of base formula. Let me try natural logarithm. Taking ln of both sides: ln(3ˣ) = ln(9). Using the logarithm power rule, that becomes x ln(3) = ln(9). Then solving for x gives x = ln(9)/ln(3). Now, ln(9) is ln(3²) which is 2 ln(3). Therefore, x = 2 ln(3) / ln(3) = 2. Yep, same answer.

So all methods lead to x being 2. Therefore, I can be confident that x equals 2 is the correct solution.

But just to make sure I didn't skip any steps or make any errors in my thinking, let me recap. Original equation: 2³·3ˣ = 72. Calculated 2³ as 8, divided both sides by 8 to get 3ˣ = 9. Recognized that 9 is 3 squared, so x is 2. Checked by plugging back in and confirmed. Also used logarithms to solve for x and still got 2.

Is there any chance I could have made a mistake in the division step? Let me verify 72 divided by 8 again. 8 times 10 is 80, which is more than 72, so 8 times 9 is 72. Yes, so 72 divided by 8 is indeed 9.

Another thing to check: Did I misread the original equation? The problem is 2³ multiplied by 3ˣ equals 72. Yes, that's what I worked with.

Alternatively, could there be another value of x that satisfies this equation? Let me think. Since 3ˣ is an exponential function, it's one-to-one, meaning that there's only one solution for x. Therefore, x = 2 is the unique solution.

So, after verifying through multiple methods and checking my steps, I'm confident that x is 2.

**Final Answer**
The value of $ x $ is \boxed{2}.

To solve the equation:

$$
2^3 \cdot 3^x = 72
$$

we follow these steps:

---

### Step 1: Simplify the known exponential term

We know that:

$$
2^3 = 8
$$

Substituting into the equation:

$$
8 \cdot 3^x = 72
$$

---

### Step 2: Isolate the exponential term

Divide both sides of the equation by 8:

$$
3^x = \frac{72}{8}
$$

$$
3^x = 9
$$

---

### Step 3: Solve for $ x $

We now solve the equation:

$$
3^x = 9
$$

Recognizing that $ 9 = 3^2 $, we can write:

$$
3^x = 3^2
$$

Since the bases are the same, we equate the exponents:

$$
x = 2
$$

---

### Step 4: Verification

Plug $ x = 2 $ back into the original equation:

$$
2^3 \cdot 3^2 = 8 \cdot 9 = 72
$$

The left-hand side equals the right-hand side, confirming our solution is correct.

---

### Alternate Method Using Logarithms

Take the natural logarithm of both sides of the simplified equation:

$$
3^x = 9
$$

$$
\ln(3^x) = \ln(9)
$$

Use the power rule of logarithms:

$$
x \ln(3) = \ln(9)
$$

$$
x = \frac{\ln(9)}{\ln(3)} = \frac{\ln(3^2)}{\ln(3)} = \frac{2 \ln(3)}{\ln(3)} = 2
$$

This confirms that $ x = 2 $ is the solution.

---

### Conclusion

The value of $ x $ that satisfies the equation is:

$$
\boxed{2}
$$

[Assistant]
Explanation:
Okay, so I need to find the value of x that makes the equation 2³ times 3ˣ equal to 72. Let me start by writing down the equation again to make sure I have it right: 2³ · 3ˣ = 72. Alright, let me break this down step by step.

First, I know that 2³ is 8 because 2 times 2 is 4, and 4 times 2 is 8. So substituting that into the equation, it becomes 8 · 3ˣ = 72. Now, I need to solve for x. Hmm, how do I isolate x here? Well, since 3ˣ is being multiplied by 8, I should probably divide both sides of the equation by 8 to get 3ˣ by itself. Let me try that.

Dividing both sides by 8 gives me 3ˣ = 72 / 8. Let me calculate 72 divided by 8. 8 times 9 is 72, so 72 divided by 8 is 9. So now the equation simplifies to 3ˣ = 9. Okay, now I need to solve for x in this equation.

Since the base here is 3 and the right side is 9, I should recall that 9 is a power of 3. Specifically, 3 squared is 9 because 3 times 3 is 9. So that means 3² = 9. Therefore, if 3ˣ equals 9, then x must be 2. Let me check my work to make sure I didn't make a mistake.

Starting with x = 2, plugging back into the original equation: 2³ · 3². 2³ is 8 and 3² is 9. Multiplying those together, 8 times 9 is indeed 72, which matches the right side of the equation. So that checks out.

Wait, but let me think again. Is there another way to approach this problem? Maybe using logarithms? Let me try that method to confirm.

Starting again with the equation 3ˣ = 9. Taking the logarithm of both sides. Since the base is 3, maybe using logarithm base 3? The logarithm base 3 of 3ˣ is x, and the logarithm base 3 of 9 is log₃(9). Since 9 is 3², log base 3 of 3² is 2. Therefore, x = 2. Same result.

Alternatively, if I use natural logarithm or common logarithm, I can apply the change of base formula. Let me try natural logarithm. Taking ln of both sides: ln(3ˣ) = ln(9). Using the logarithm power rule, that becomes x ln(3) = ln(9). Then solving for x gives x = ln(9)/ln(3). Now, ln(9) is ln(3²) which is 2 ln(3). Therefore, x = 2 ln(3) / ln(3) = 2. Yep, same answer.

So all methods lead to x being 2. Therefore, I can be confident that x equals 2 is the correct solution.

But just to make sure I didn't skip any steps or make any errors in my thinking, let me recap. Original equation: 2³·3ˣ = 72. Calculated 2³ as 8, divided both sides by 8 to get 3ˣ = 9. Recognized that 9 is 3 squared, so x is 2. Checked by plugging back in and confirmed. Also used logarithms to solve for x and still got 2.

Is there any chance I could have made a mistake in the division step? Let me verify 72 divided by 8 again. 8 times 10 is 80, which is more than 72, so 8 times 9 is 72. Yes, so 72 divided by 8 is indeed 9.

Another thing to check: Did I misread the original equation? The problem is 2³ multiplied by 3ˣ equals 72. Yes, that's what I worked with.

Alternatively, could there be another value of x that satisfies this equation? Let me think. Since 3ˣ is an exponential function, it's one-to-one, meaning that there's only one solution for x. Therefore, x = 2 is the unique solution.

So, after verifying through multiple methods and checking my steps, I'm confident that x is 2.

**Final Answer**
The value of $ x $ is \boxed{2}.

To solve the equation:

$$
2^3 \cdot 3^x = 72
$$

we follow these steps:

---

### Step 1: Simplify the known exponential term

We know that:

$$
2^3 = 8
$$

Substituting into the equation:

$$
8 \cdot 3^x = 72
$$

---

### Step 2: Isolate the exponential term

Divide both sides of the equation by 8:

$$
3^x = \frac{72}{8}
$$

$$
3^x = 9
$$

---

### Step 3: Solve for $ x $

We now solve the equation:

$$
3^x = 9
$$

Recognizing that $ 9 = 3^2 $, we can write:

$$
3^x = 3^2
$$

Since the bases are the same, we equate the exponents:

$$
x = 2
$$

---

### Step 4: Verification

Plug $ x = 2 $ back into the original equation:

$$
2^3 \cdot 3^2 = 8 \cdot 9 = 72
$$

The left-hand side equals the right-hand side, confirming our solution is correct.

---

### Alternate Method Using Logarithms

Take the natural logarithm of both sides of the simplified equation:

$$
3^x = 9
$$

$$
\ln(3^x) = \ln(9)
$$

Use the power rule of logarithms:

$$
x \ln(3) = \ln(9)
$$

$$
x = \frac{\ln(9)}{\ln(3)} = \frac{\ln(3^2)}{\ln(3)} = \frac{2 \ln(3)}{\ln(3)} = 2
$$

This confirms that $ x = 2 $ is the solution.

---

### Conclusion

The value of $ x $ that satisfies the equation is:

$$
\boxed{2}
$$

Solution: Since the prime factorization of 72 is $72=2^3\cdot 3^2$, we have $x=\boxed{2}$.


Explanation:
Okay, so I need to figure out how many different outfits I can make with the given clothes. Let me see... I have 5 shirts, 6 pairs of pants, and 8 hats. The question says each outfit has to consist of one shirt, one pair of pants, and one hat. Hmm, right. So I guess I need to combine these items in different ways.

Let me start by thinking about combinations. If I have multiple choices for each item, then the total number of combinations should be the product of the number of choices for each item. Wait, is that right? Like, for each shirt, I can pair it with any pants and any hat? Yeah, that seems to make sense.

So, let me break it down. For each shirt, I can choose any of the 6 pairs of pants. So if I have 5 shirts, then for each shirt, there are 6 pants options. So that would be 5 multiplied by 6. But then, also, for each of those shirt-pants combinations, I can choose any of the 8 hats. So then I need to multiply that result by 8.

Let me write that down step by step. First, shirts and pants: 5 shirts * 6 pants = 30 combinations. Then, for each of those 30 combinations, there are 8 hats. So 30 * 8 = 240. So the total number of outfits would be 240?

Wait, but let me check if there's another way to think about it. Maybe using the multiplication principle? The multiplication principle says that if there are n ways to do something and m ways to do another thing independently, then there are n*m ways to do both.

So here, choosing a shirt, a pair of pants, and a hat are independent choices. So the number of ways to choose all three is the product of the number of choices for each. So that would be 5 * 6 * 8. Let me compute that.

First multiply 5 and 6: 5*6=30. Then 30*8=240. Yep, same answer. So that seems to confirm it.

But let me make sure I didn't miss anything. The problem says "outfits consisting of one shirt, one pair of pants, and one hat." So each outfit has exactly one of each. There's no mention of optional items or anything else. So I don't need to worry about combinations where maybe you have multiple shirts or something.

Also, are there any restrictions? Like, maybe some shirts don't go with certain pants? But the problem doesn't mention any such restrictions. So we can assume that any shirt can be worn with any pants and any hat.

Therefore, the total number of possible outfits is indeed 5*6*8=240.

Wait, just to make sure, let me think of a smaller example. Suppose I have 2 shirts, 3 pants, and 4 hats. Then according to this logic, the number of outfits would be 2*3*4=24. Let me count manually. For each shirt, there are 3 pants, so 2*3=6 shirt-pants combinations. Then for each of those, 4 hats, so 6*4=24. That seems correct.

Alternatively, if I list them: Shirt 1 with pants 1, 2, 3 and hats 1,2,3,4: that's 3*4=12. Similarly, Shirt 2 with pants 1,2,3 and hats 1,2,3,4: another 12. Total 24. Yeah, that's correct.

Therefore, going back to the original problem, 5*6*8=240 should be correct.

I think that's all. I don't see any other factors or possible errors in the reasoning. So I think the answer is 240.

**Final Answer**
The number of possible outfits is \boxed{240}.

To determine the total number of possible outfits, we analyze the problem by considering each component independently and then combining them.

---

### Step 1: Understand the Choices

We are given:

- **5 shirts**
- **6 pairs of pants**
- **8 hats**

Each outfit must consist of **exactly one** of each item: one shirt, one pair of pants, and one hat.

---

### Step 2: Apply the Multiplication Principle

The **Multiplication Principle** states that if there are `n` ways to do one thing and `m` ways to do another, and the choices are independent, then the number of total ways to do both is:

$$
n \times m
$$

In this case:

- There are 5 choices for shirts.
- For each shirt, there are 6 choices for pants.
- For each shirt and pants combination, there are 8 choices for hats.

So, the total number of combinations is:

$$
5 \times 6 \times 8
$$

---

### Step 3: Perform the Calculation

First, compute the product of shirts and pants:

$$
5 \times 6 = 30
$$

Then multiply by the number of hats:

$$
30 \times 8 = 240
$$

---

### Final Result

Therefore, the total number of different outfits that can be made is:

$$
\boxed{240}
$$

[Assistant]
Explanation:
Okay, so I need to figure out how many different outfits I can make with the given clothes. Let me see... I have 5 shirts, 6 pairs of pants, and 8 hats. The question says each outfit has to consist of one shirt, one pair of pants, and one hat. Hmm, right. So I guess I need to combine these items in different ways.

Let me start by thinking about combinations. If I have multiple choices for each item, then the total number of combinations should be the product of the number of choices for each item. Wait, is that right? Like, for each shirt, I can pair it with any pants and any hat? Yeah, that seems to make sense.

So, let me break it down. For each shirt, I can choose any of the 6 pairs of pants. So if I have 5 shirts, then for each shirt, there are 6 pants options. So that would be 5 multiplied by 6. But then, also, for each of those shirt-pants combinations, I can choose any of the 8 hats. So then I need to multiply that result by 8.

Let me write that down step by step. First, shirts and pants: 5 shirts * 6 pants = 30 combinations. Then, for each of those 30 combinations, there are 8 hats. So 30 * 8 = 240. So the total number of outfits would be 240?

Wait, but let me check if there's another way to think about it. Maybe using the multiplication principle? The multiplication principle says that if there are n ways to do something and m ways to do another thing independently, then there are n*m ways to do both.

So here, choosing a shirt, a pair of pants, and a hat are independent choices. So the number of ways to choose all three is the product of the number of choices for each. So that would be 5 * 6 * 8. Let me compute that.

First multiply 5 and 6: 5*6=30. Then 30*8=240. Yep, same answer. So that seems to confirm it.

But let me make sure I didn't miss anything. The problem says "outfits consisting of one shirt, one pair of pants, and one hat." So each outfit has exactly one of each. There's no mention of optional items or anything else. So I don't need to worry about combinations where maybe you have multiple shirts or something.

Also, are there any restrictions? Like, maybe some shirts don't go with certain pants? But the problem doesn't mention any such restrictions. So we can assume that any shirt can be worn with any pants and any hat.

Therefore, the total number of possible outfits is indeed 5*6*8=240.

Wait, just to make sure, let me think of a smaller example. Suppose I have 2 shirts, 3 pants, and 4 hats. Then according to this logic, the number of outfits would be 2*3*4=24. Let me count manually. For each shirt, there are 3 pants, so 2*3=6 shirt-pants combinations. Then for each of those, 4 hats, so 6*4=24. That seems correct.

Alternatively, if I list them: Shirt 1 with pants 1, 2, 3 and hats 1,2,3,4: that's 3*4=12. Similarly, Shirt 2 with pants 1,2,3 and hats 1,2,3,4: another 12. Total 24. Yeah, that's correct.

Therefore, going back to the original problem, 5*6*8=240 should be correct.

I think that's all. I don't see any other factors or possible errors in the reasoning. So I think the answer is 240.

**Final Answer**
The number of possible outfits is \boxed{240}.

To determine the total number of possible outfits, we analyze the problem by considering each component independently and then combining them.

---

### Step 1: Understand the Choices

We are given:

- **5 shirts**
- **6 pairs of pants**
- **8 hats**

Each outfit must consist of **exactly one** of each item: one shirt, one pair of pants, and one hat.

---

### Step 2: Apply the Multiplication Principle

The **Multiplication Principle** states that if there are `n` ways to do one thing and `m` ways to do another, and the choices are independent, then the number of total ways to do both is:

$$
n \times m
$$

In this case:

- There are 5 choices for shirts.
- For each shirt, there are 6 choices for pants.
- For each shirt and pants combination, there are 8 choices for hats.

So, the total number of combinations is:

$$
5 \times 6 \times 8
$$

---

### Step 3: Perform the Calculation

First, compute the product of shirts and pants:

$$
5 \times 6 = 30
$$

Then multiply by the number of hats:

$$
30 \times 8 = 240
$$

---

### Final Result

Therefore, the total number of different outfits that can be made is:

$$
\boxed{240}
$$

Solution: There are 5 options for shirts, 6 options for pants, and 8 options for hats, for a total of $5 \times 6 \times 8 = \boxed{240}$ outfits.


Explanation:
Okay, so I need to solve the equation |5x - 1| = x + 3 and find the largest value of x that satisfies it. Hmm, let me think about how to approach absolute value equations. I remember that absolute value equations can have two cases because the expression inside the absolute value can be positive or negative. So, maybe I should split this into two separate equations and solve each one?

First, let me recall the definition of absolute value. The absolute value of a number is its distance from zero on the number line, so |a| = a if a ≥ 0, and |a| = -a if a < 0. Therefore, for the equation |5x - 1| = x + 3, I need to consider two cases:

Case 1: 5x - 1 ≥ 0, which simplifies to 5x ≥ 1, so x ≥ 1/5. In this case, |5x - 1| = 5x - 1, so the equation becomes 5x - 1 = x + 3.

Case 2: 5x - 1 < 0, which means 5x < 1, so x < 1/5. In this case, |5x - 1| = -(5x - 1) = -5x + 1, so the equation becomes -5x + 1 = x + 3.

Okay, that seems right. Now I need to solve both equations and check the solutions in the original equation because sometimes when dealing with absolute values, you can get extraneous solutions.

Starting with Case 1: 5x - 1 = x + 3.

Let me solve for x. Subtract x from both sides: 5x - x - 1 = 3 ⇒ 4x - 1 = 3. Then add 1 to both sides: 4x = 4. Divide by 4: x = 1.

Now, I need to check if this solution satisfies the condition of Case 1, which is x ≥ 1/5. Since 1 is definitely greater than or equal to 1/5, this solution is valid. So x = 1 is a valid solution.

Now moving on to Case 2: -5x + 1 = x + 3.

Let me solve this equation. Let me get all the x terms on one side. Let's add 5x to both sides: 1 = 6x + 3. Then subtract 3 from both sides: 1 - 3 = 6x ⇒ -2 = 6x. Divide both sides by 6: x = -2/6 = -1/3.

Now, check if this solution satisfies the condition of Case 2, which is x < 1/5. Since -1/3 is approximately -0.333..., which is certainly less than 1/5 (which is 0.2), this solution is also valid. So x = -1/3 is another valid solution.

Wait, so the solutions are x = 1 and x = -1/3? Let me verify both in the original equation to make sure.

First, check x = 1: |5(1) - 1| = |5 - 1| = |4| = 4. On the right side, x + 3 = 1 + 3 = 4. So 4 = 4, which works.

Now check x = -1/3: |5(-1/3) - 1| = |-5/3 - 1| = |-5/3 - 3/3| = |-8/3| = 8/3. On the right side, x + 3 = -1/3 + 3 = -1/3 + 9/3 = 8/3. So 8/3 = 8/3, which works too. Great, both solutions are valid.

So the equation has two solutions: x = -1/3 and x = 1. The problem asks for the largest value of x that satisfies the equation, so between -1/3 and 1, the larger one is 1. Therefore, the answer should be 1. But wait, let me make sure there are no other solutions or if I made any mistakes in my reasoning.

Wait, could there be more cases? No, because absolute value equations typically split into two cases based on the expression inside the absolute value being positive or negative. So I think that covers all possibilities.

Alternatively, another method to solve absolute value equations is to square both sides to eliminate the absolute value, but that can sometimes introduce extraneous solutions. Let me try that approach as a check.

Starting with |5x - 1| = x + 3. Squaring both sides:

(5x - 1)^2 = (x + 3)^2.

Expanding both sides:

Left side: 25x² - 10x + 1.

Right side: x² + 6x + 9.

Bring all terms to one side:

25x² - 10x + 1 - x² - 6x - 9 = 0 ⇒ 24x² - 16x - 8 = 0.

Simplify by dividing all terms by 8:

3x² - 2x - 1 = 0.

Now, solving this quadratic equation. Using the quadratic formula: x = [2 ± √(4 + 12)] / 6 = [2 ± √16]/6 = [2 ± 4]/6.

So two solutions:

x = (2 + 4)/6 = 6/6 = 1.

x = (2 - 4)/6 = (-2)/6 = -1/3.

Same solutions as before! So that confirms that x = 1 and x = -1/3 are the solutions. Therefore, the largest one is 1.

But just to be thorough, when squaring both sides of an equation, sometimes you can introduce solutions that don't satisfy the original equation. So even though we found the same solutions, we still need to check them in the original equation, which we already did. So that's good.

Alternatively, another way to think about the original equation is to consider the graph of both sides. The left side is an absolute value function, which V-shaped, and the right side is a linear function. The solutions to the equation are the points where these two graphs intersect. But since we already solved it algebraically and verified, this might not be necessary, but just another way to visualize.

But just to think, the absolute value function |5x - 1| has a vertex at x = 1/5, since the expression inside is 5x - 1, so the vertex is where 5x - 1 = 0, which is x = 1/5. The right side is a line with slope 1 and y-intercept 3. So depending on the intersection points, which we found at x = -1/3 and x = 1, which makes sense.

So, all checks out. Therefore, the largest value of x is 1. Therefore, the answer is 1.

But wait, just to make sure I didn't miss anything. Let me think again. Suppose someone might think that when you split into cases, maybe there are more cases? For example, sometimes when dealing with absolute values, people might consider different intervals. Wait, but in this case, the expression inside the absolute value is 5x - 1. So the critical point is at x = 1/5. Therefore, the two cases are x ≥ 1/5 and x < 1/5. So that's all the cases. So I think that's covered.

Alternatively, someone might forget to check if the right side of the original equation is non-negative. Because absolute value is always non-negative, so the right side must also be non-negative. Therefore, x + 3 ≥ 0 ⇒ x ≥ -3. So even though our solutions are x = -1/3 and x = 1, both of which are greater than -3, so that's okay. If we had solutions less than -3, they would be extraneous. But in our case, both solutions are valid. So that's another check.

So, for example, if we had a solution x = -4, we would have to reject it because the right side would be -4 + 3 = -1, which is negative, but the left side is absolute value, which can't be negative. But our solutions are okay. So that's another check.

So all in all, my conclusion is that the largest value of x is 1. Therefore, the answer is \boxed{1}.

**Final Answer**
The largest value of $ x $ that satisfies the equation is \boxed{1}.

To solve the equation:

$$
|5x - 1| = x + 3
$$

we begin by analyzing the absolute value on the left-hand side. The absolute value of an expression is either:

- **Case 1:** The expression itself when it is non-negative, or
- **Case 2:** The negative of the expression when it is negative.

---

### **Case 1: $5x - 1 \geq 0 \Rightarrow x \geq \dfrac{1}{5}$**

If $5x - 1 \geq 0$, then:

$$
|5x - 1| = 5x - 1
$$

So the equation becomes:

$$
5x - 1 = x + 3
$$

Subtract $x$ from both sides:

$$
4x - 1 = 3
$$

Add 1 to both sides:

$$
4x = 4
$$

Divide by 4:

$$
x = 1
$$

Now, check if this solution lies within the condition $x \geq \frac{1}{5}$, which it does. So this is a **valid** solution.

---

### **Case 2: $5x - 1 < 0 \Rightarrow x < \dfrac{1}{5}$**

If $5x - 1 < 0$, then:

$$
|5x - 1| = -(5x - 1) = -5x + 1
$$

So the equation becomes:

$$
-5x + 1 = x + 3
$$

Subtract $x$ from both sides:

$$
-6x + 1 = 3
$$

Subtract 1 from both sides:

$$
-6x = 2
$$

Divide by -6:

$$
x = -\dfrac{1}{3}
$$

Now, check if $x = -\dfrac{1}{3}$ is less than $\dfrac{1}{5}$. Yes, it is. So this is also a **valid** solution.

---

### **Verification of Both Solutions**

We should verify both solutions in the original equation to ensure they satisfy it.

- For $x = 1$:

$$
|5(1) - 1| = |4| = 4 \quad \text{and} \quad 1 + 3 = 4 \quad \Rightarrow \text{Valid}
$$

- For $x = -\dfrac{1}{3}$:

$$
|5(-\dfrac{1}{3}) - 1| = |-\dfrac{5}{3} - 1| = |-\dfrac{8}{3}| = \dfrac{8}{3}
$$

$$
x + 3 = -\dfrac{1}{3} + 3 = \dfrac{8}{3} \quad \Rightarrow \text{Valid}
$$

---

### **Conclusion**

The two solutions to the equation are:

$$
x = 1 \quad \text{and} \quad x = -\dfrac{1}{3}
$$

Between these, the **larger** value is:

$$
\boxed{1}
$$

[Assistant]
Explanation:
Okay, so I need to solve the equation |5x - 1| = x + 3 and find the largest value of x that satisfies it. Hmm, let me think about how to approach absolute value equations. I remember that absolute value equations can have two cases because the expression inside the absolute value can be positive or negative. So, maybe I should split this into two separate equations and solve each one?

First, let me recall the definition of absolute value. The absolute value of a number is its distance from zero on the number line, so |a| = a if a ≥ 0, and |a| = -a if a < 0. Therefore, for the equation |5x - 1| = x + 3, I need to consider two cases:

Case 1: 5x - 1 ≥ 0, which simplifies to 5x ≥ 1, so x ≥ 1/5. In this case, |5x - 1| = 5x - 1, so the equation becomes 5x - 1 = x + 3.

Case 2: 5x - 1 < 0, which means 5x < 1, so x < 1/5. In this case, |5x - 1| = -(5x - 1) = -5x + 1, so the equation becomes -5x + 1 = x + 3.

Okay, that seems right. Now I need to solve both equations and check the solutions in the original equation because sometimes when dealing with absolute values, you can get extraneous solutions.

Starting with Case 1: 5x - 1 = x + 3.

Let me solve for x. Subtract x from both sides: 5x - x - 1 = 3 ⇒ 4x - 1 = 3. Then add 1 to both sides: 4x = 4. Divide by 4: x = 1.

Now, I need to check if this solution satisfies the condition of Case 1, which is x ≥ 1/5. Since 1 is definitely greater than or equal to 1/5, this solution is valid. So x = 1 is a valid solution.

Now moving on to Case 2: -5x + 1 = x + 3.

Let me solve this equation. Let me get all the x terms on one side. Let's add 5x to both sides: 1 = 6x + 3. Then subtract 3 from both sides: 1 - 3 = 6x ⇒ -2 = 6x. Divide both sides by 6: x = -2/6 = -1/3.

Now, check if this solution satisfies the condition of Case 2, which is x < 1/5. Since -1/3 is approximately -0.333..., which is certainly less than 1/5 (which is 0.2), this solution is also valid. So x = -1/3 is another valid solution.

Wait, so the solutions are x = 1 and x = -1/3? Let me verify both in the original equation to make sure.

First, check x = 1: |5(1) - 1| = |5 - 1| = |4| = 4. On the right side, x + 3 = 1 + 3 = 4. So 4 = 4, which works.

Now check x = -1/3: |5(-1/3) - 1| = |-5/3 - 1| = |-5/3 - 3/3| = |-8/3| = 8/3. On the right side, x + 3 = -1/3 + 3 = -1/3 + 9/3 = 8/3. So 8/3 = 8/3, which works too. Great, both solutions are valid.

So the equation has two solutions: x = -1/3 and x = 1. The problem asks for the largest value of x that satisfies the equation, so between -1/3 and 1, the larger one is 1. Therefore, the answer should be 1. But wait, let me make sure there are no other solutions or if I made any mistakes in my reasoning.

Wait, could there be more cases? No, because absolute value equations typically split into two cases based on the expression inside the absolute value being positive or negative. So I think that covers all possibilities.

Alternatively, another method to solve absolute value equations is to square both sides to eliminate the absolute value, but that can sometimes introduce extraneous solutions. Let me try that approach as a check.

Starting with |5x - 1| = x + 3. Squaring both sides:

(5x - 1)^2 = (x + 3)^2.

Expanding both sides:

Left side: 25x² - 10x + 1.

Right side: x² + 6x + 9.

Bring all terms to one side:

25x² - 10x + 1 - x² - 6x - 9 = 0 ⇒ 24x² - 16x - 8 = 0.

Simplify by dividing all terms by 8:

3x² - 2x - 1 = 0.

Now, solving this quadratic equation. Using the quadratic formula: x = [2 ± √(4 + 12)] / 6 = [2 ± √16]/6 = [2 ± 4]/6.

So two solutions:

x = (2 + 4)/6 = 6/6 = 1.

x = (2 - 4)/6 = (-2)/6 = -1/3.

Same solutions as before! So that confirms that x = 1 and x = -1/3 are the solutions. Therefore, the largest one is 1.

But just to be thorough, when squaring both sides of an equation, sometimes you can introduce solutions that don't satisfy the original equation. So even though we found the same solutions, we still need to check them in the original equation, which we already did. So that's good.

Alternatively, another way to think about the original equation is to consider the graph of both sides. The left side is an absolute value function, which V-shaped, and the right side is a linear function. The solutions to the equation are the points where these two graphs intersect. But since we already solved it algebraically and verified, this might not be necessary, but just another way to visualize.

But just to think, the absolute value function |5x - 1| has a vertex at x = 1/5, since the expression inside is 5x - 1, so the vertex is where 5x - 1 = 0, which is x = 1/5. The right side is a line with slope 1 and y-intercept 3. So depending on the intersection points, which we found at x = -1/3 and x = 1, which makes sense.

So, all checks out. Therefore, the largest value of x is 1. Therefore, the answer is 1.

But wait, just to make sure I didn't miss anything. Let me think again. Suppose someone might think that when you split into cases, maybe there are more cases? For example, sometimes when dealing with absolute values, people might consider different intervals. Wait, but in this case, the expression inside the absolute value is 5x - 1. So the critical point is at x = 1/5. Therefore, the two cases are x ≥ 1/5 and x < 1/5. So that's all the cases. So I think that's covered.

Alternatively, someone might forget to check if the right side of the original equation is non-negative. Because absolute value is always non-negative, so the right side must also be non-negative. Therefore, x + 3 ≥ 0 ⇒ x ≥ -3. So even though our solutions are x = -1/3 and x = 1, both of which are greater than -3, so that's okay. If we had solutions less than -3, they would be extraneous. But in our case, both solutions are valid. So that's another check.

So, for example, if we had a solution x = -4, we would have to reject it because the right side would be -4 + 3 = -1, which is negative, but the left side is absolute value, which can't be negative. But our solutions are okay. So that's another check.

So all in all, my conclusion is that the largest value of x is 1. Therefore, the answer is \boxed{1}.

**Final Answer**
The largest value of $ x $ that satisfies the equation is \boxed{1}.

To solve the equation:

$$
|5x - 1| = x + 3
$$

we begin by analyzing the absolute value on the left-hand side. The absolute value of an expression is either:

- **Case 1:** The expression itself when it is non-negative, or
- **Case 2:** The negative of the expression when it is negative.

---

### **Case 1: $5x - 1 \geq 0 \Rightarrow x \geq \dfrac{1}{5}$**

If $5x - 1 \geq 0$, then:

$$
|5x - 1| = 5x - 1
$$

So the equation becomes:

$$
5x - 1 = x + 3
$$

Subtract $x$ from both sides:

$$
4x - 1 = 3
$$

Add 1 to both sides:

$$
4x = 4
$$

Divide by 4:

$$
x = 1
$$

Now, check if this solution lies within the condition $x \geq \frac{1}{5}$, which it does. So this is a **valid** solution.

---

### **Case 2: $5x - 1 < 0 \Rightarrow x < \dfrac{1}{5}$**

If $5x - 1 < 0$, then:

$$
|5x - 1| = -(5x - 1) = -5x + 1
$$

So the equation becomes:

$$
-5x + 1 = x + 3
$$

Subtract $x$ from both sides:

$$
-6x + 1 = 3
$$

Subtract 1 from both sides:

$$
-6x = 2
$$

Divide by -6:

$$
x = -\dfrac{1}{3}
$$

Now, check if $x = -\dfrac{1}{3}$ is less than $\dfrac{1}{5}$. Yes, it is. So this is also a **valid** solution.

---

### **Verification of Both Solutions**

We should verify both solutions in the original equation to ensure they satisfy it.

- For $x = 1$:

$$
|5(1) - 1| = |4| = 4 \quad \text{and} \quad 1 + 3 = 4 \quad \Rightarrow \text{Valid}
$$

- For $x = -\dfrac{1}{3}$:

$$
|5(-\dfrac{1}{3}) - 1| = |-\dfrac{5}{3} - 1| = |-\dfrac{8}{3}| = \dfrac{8}{3}
$$

$$
x + 3 = -\dfrac{1}{3} + 3 = \dfrac{8}{3} \quad \Rightarrow \text{Valid}
$$

---

### **Conclusion**

The two solutions to the equation are:

$$
x = 1 \quad \text{and} \quad x = -\dfrac{1}{3}
$$

Between these, the **larger** value is:

$$
\boxed{1}
$$

Solution:
We can split the expression $|5x-1|=x+3$ into two separate cases. In the first case, \begin{align*} 5x-1&=x+3
\\\Rightarrow \qquad 4x&=4
\\\Rightarrow \qquad x&=1
\end{align*}If we plug this value of $x$ back into the original equation in order to check our answer, we see that $|5(1)-1|=1+3$ or $4=4$. Since this is true, we can accept $x=1$ as a valid solution.

In the second case, \begin{align*} 5x-1&=-(x+3)
\\ 5x-1&=-x-3
\\\Rightarrow \qquad 6x&=-2
\\\Rightarrow \qquad x&=-\frac13.
\end{align*}If we plug $-\frac13$ back into the initial equation, we get that $\left|5\left(-\frac13\right)-1\right|=-\frac13+3$, which simplifies to $\left|-\frac{8}{3}\right|=\frac{8}{3}$, or $\frac{8}{3}=\frac{8}{3}$. Since this is also a true statement, we can accept $x=-\frac13$ as a valid solution as well. Since both $x=1$ and $x=-\frac13$ are possible values of $x$, the final answer is the larger of the two: $x=\boxed{1}$.