Explanation:
Okay, so I need to figure out the bicycle's speed in inches per second when it's given as 20 feet per minute. Let me think about how to convert the units here.

First, I know that feet and inches are both units of length, and minutes and seconds are units of time. So I need to convert feet to inches and minutes to seconds. Let me recall the conversion factors.

There are 12 inches in a foot, right? So if the bicycle is moving at 20 feet per minute, that's equivalent to 20 times 12 inches per minute. Let me write that down:

20 feet/minute * 12 inches/foot = 240 inches/minute.

Okay, so that converts the distance part from feet to inches. Now, the time unit is minutes, and I need to convert that to seconds. Since there are 60 seconds in a minute, I can divide the number of inches per minute by 60 to get inches per second.

So taking the 240 inches per minute and dividing by 60 seconds per minute:

240 inches/minute ÷ 60 seconds/minute = 4 inches/second.

Wait, let me check that again. If I have 240 inches in a minute, then how many inches per second is that? Since one minute is 60 seconds, dividing by 60 would give me the number of inches per second.

Yes, 240 divided by 60 is 4. So that seems straightforward.

But let me make sure I didn't mix up any conversion factors. Let me go through it step by step again.

Original speed: 20 feet per minute.

Convert feet to inches: 1 foot = 12 inches. So 20 feet is 20 * 12 inches. Let me calculate that again. 20 * 12 is 240. Yep, that's correct.

So 20 feet per minute is 240 inches per minute.

Now, converting minutes to seconds. There are 60 seconds in a minute, so to get inches per second, divide by 60.

240 inches per minute divided by 60 seconds per minute equals 4 inches per second.

Hmm, that seems right. Let me think if there's another way to approach this. Maybe using dimensional analysis?

Starting with 20 feet/minute. Multiply by inches per foot to get inches per minute, then multiply by minutes per second to get inches per second.

So:

20 feet/minute * (12 inches / 1 foot) * (1 minute / 60 seconds) = ?

The feet unit cancels out, the minute unit cancels out, leaving inches per second.

Calculating the numbers: 20 * 12 = 240, then 240 / 60 = 4. Yep, same result.

So that seems to confirm it.

Alternatively, maybe convert the entire thing to inches per second directly. Let me think.

If 1 foot is 12 inches, and 1 minute is 60 seconds, then 20 feet per minute is equal to 20 * 12 inches per 60 seconds. Which is (20 * 12)/60 inches per second.

Which is again 240 / 60 = 4.

So all methods lead to 4 inches per second.

Wait, but just to make sure I didn't make a mistake in arithmetic. Let me check 20 times 12. 20*12 is 240. Correct. 240 divided by 60 is 4. Correct.

Is there any chance I confused feet with something else? Like, maybe the question was about something else? No, the question says a bicycle is traveling at 20 feet per minute, and asks for inches per second.

Another way to think about it: How many inches does the bicycle travel in one second?

If it's going 20 feet per minute, then in one second, it's going 20 feet per 60 seconds. So per second, it's 20/60 feet per second. Then convert that to inches.

So 20/60 feet per second is equal to (20/60)*12 inches per second.

Calculating that: 20 divided by 60 is 1/3. 1/3 * 12 is 4.

Same answer.

So regardless of the method, it's 4 inches per second.

Therefore, I think the answer is 4 inches per second.

**Final Answer**
The bicycle's speed is \boxed{4} inches per second.

To determine the bicycle's speed in inches per second, we begin with the given speed of **20 feet per minute** and convert it to the desired units using appropriate conversion factors.

---

### Step 1: Convert Feet to Inches

We know that:

$$
1 \text{ foot} = 12 \text{ inches}
$$

So, to convert 20 feet per minute to inches per minute:

$$
20 \text{ feet/minute} \times 12 \text{ inches/foot} = 240 \text{ inches/minute}
$$

---

### Step 2: Convert Minutes to Seconds

There are 60 seconds in a minute. To convert inches per minute to inches per second, divide by 60:

$$
240 \text{ inches/minute} \div 60 \text{ seconds/minute} = 4 \text{ inches/second}
$$

---

### Final Answer

$$
\boxed{4}
$$

The bicycle's speed is **4 inches per second**.

[Assistant]
Explanation:
Okay, so I need to figure out the bicycle's speed in inches per second when it's given as 20 feet per minute. Let me think about how to convert the units here.

First, I know that feet and inches are both units of length, and minutes and seconds are units of time. So I need to convert feet to inches and minutes to seconds. Let me recall the conversion factors.

There are 12 inches in a foot, right? So if the bicycle is moving at 20 feet per minute, that's equivalent to 20 times 12 inches per minute. Let me write that down:

20 feet/minute * 12 inches/foot = 240 inches/minute.

Okay, so that converts the distance part from feet to inches. Now, the time unit is minutes, and I need to convert that to seconds. Since there are 60 seconds in a minute, I can divide the number of inches per minute by 60 to get inches per second.

So taking the 240 inches per minute and dividing by 60 seconds per minute:

240 inches/minute ÷ 60 seconds/minute = 4 inches/second.

Wait, let me check that again. If I have 240 inches in a minute, then how many inches per second is that? Since one minute is 60 seconds, dividing by 60 would give me the number of inches per second.

Yes, 240 divided by 60 is 4. So that seems straightforward.

But let me make sure I didn't mix up any conversion factors. Let me go through it step by step again.

Original speed: 20 feet per minute.

Convert feet to inches: 1 foot = 12 inches. So 20 feet is 20 * 12 inches. Let me calculate that again. 20 * 12 is 240. Yep, that's correct.

So 20 feet per minute is 240 inches per minute.

Now, converting minutes to seconds. There are 60 seconds in a minute, so to get inches per second, divide by 60.

240 inches per minute divided by 60 seconds per minute equals 4 inches per second.

Hmm, that seems right. Let me think if there's another way to approach this. Maybe using dimensional analysis?

Starting with 20 feet/minute. Multiply by inches per foot to get inches per minute, then multiply by minutes per second to get inches per second.

So:

20 feet/minute * (12 inches / 1 foot) * (1 minute / 60 seconds) = ?

The feet unit cancels out, the minute unit cancels out, leaving inches per second.

Calculating the numbers: 20 * 12 = 240, then 240 / 60 = 4. Yep, same result.

So that seems to confirm it.

Alternatively, maybe convert the entire thing to inches per second directly. Let me think.

If 1 foot is 12 inches, and 1 minute is 60 seconds, then 20 feet per minute is equal to 20 * 12 inches per 60 seconds. Which is (20 * 12)/60 inches per second.

Which is again 240 / 60 = 4.

So all methods lead to 4 inches per second.

Wait, but just to make sure I didn't make a mistake in arithmetic. Let me check 20 times 12. 20*12 is 240. Correct. 240 divided by 60 is 4. Correct.

Is there any chance I confused feet with something else? Like, maybe the question was about something else? No, the question says a bicycle is traveling at 20 feet per minute, and asks for inches per second.

Another way to think about it: How many inches does the bicycle travel in one second?

If it's going 20 feet per minute, then in one second, it's going 20 feet per 60 seconds. So per second, it's 20/60 feet per second. Then convert that to inches.

So 20/60 feet per second is equal to (20/60)*12 inches per second.

Calculating that: 20 divided by 60 is 1/3. 1/3 * 12 is 4.

Same answer.

So regardless of the method, it's 4 inches per second.

Therefore, I think the answer is 4 inches per second.

**Final Answer**
The bicycle's speed is \boxed{4} inches per second.

To determine the bicycle's speed in inches per second, we begin with the given speed of **20 feet per minute** and convert it to the desired units using appropriate conversion factors.

---

### Step 1: Convert Feet to Inches

We know that:

$$
1 \text{ foot} = 12 \text{ inches}
$$

So, to convert 20 feet per minute to inches per minute:

$$
20 \text{ feet/minute} \times 12 \text{ inches/foot} = 240 \text{ inches/minute}
$$

---

### Step 2: Convert Minutes to Seconds

There are 60 seconds in a minute. To convert inches per minute to inches per second, divide by 60:

$$
240 \text{ inches/minute} \div 60 \text{ seconds/minute} = 4 \text{ inches/second}
$$

---

### Final Answer

$$
\boxed{4}
$$

The bicycle's speed is **4 inches per second**.

Solution: There are $12$ inches in a foot, so the bicycle is traveling at $12(20)=240$ inches per minute.  There are $60$ seconds in a minute, so the bicycle is traveling at $\frac{240}{60}=\boxed{4}$ inches per second.


Explanation:
Okay, so I need to find the minimum value of (x + y)(y + z) given that x, y, z are positive real numbers and satisfy the condition xyz(x + y + z) = 1. Hmm, let me think about how to approach this.

First, I remember that optimization problems with constraints often use methods like Lagrange multipliers, but since this is likely a problem from algebra or inequalities, maybe I can use AM-GM inequality or some substitution. Let me try to recall the AM-GM inequality. It states that for non-negative numbers, the arithmetic mean is greater than or equal to the geometric mean. Maybe I can apply that here.

But before jumping into inequalities, let me write down what I need to minimize and the constraint:

Minimize (x + y)(y + z) subject to xyz(x + y + z) = 1.

Let me expand (x + y)(y + z) to see if that helps. Expanding it:

(x + y)(y + z) = xy + xz + y² + yz.

Hmm, not sure if that helps directly. Maybe I can think of variables substitution. Let me consider variables a, b, c or something else. Alternatively, maybe express variables in terms of each other.

Alternatively, since there are three variables and one constraint, maybe I can reduce the number of variables. Let me think. Let me try to set variables such that the constraint becomes simpler. For example, maybe set x = z? Because the expression (x + y)(y + z) is symmetric in x and z. If I assume x = z, maybe that would simplify things. Let me try that.

Let me assume x = z. Then, the constraint becomes x * y * x * (x + y + x) = x² y (2x + y) = 1. And the expression to minimize becomes (x + y)(y + x) = (x + y)². So, if I can find the minimum of (x + y)² given that x² y (2x + y) = 1. Maybe this is easier?

Alternatively, maybe not. Let me check if assuming x = z is valid. Since the original problem is symmetric in x and z, maybe the minimum occurs when x = z. But I need to verify that. However, since I don't know yet, maybe I should proceed with this assumption and see if I can find a solution, and then check if it's indeed the minimum.

So, assuming x = z, then we have:

Let me denote x = z. Let me set variables as x, y, x. Then, the constraint is x² y (2x + y) = 1. Let me denote t = y/x. Let me set t as a positive real number. Then y = t x. Then substituting into the constraint:

x² * (t x) * (2x + t x) = x² * t x * x (2 + t) = t (2 + t) x^4 = 1. Therefore, x^4 = 1 / [t(2 + t)].

So x = [1 / (t(2 + t))]^{1/4}.

Then, the expression to minimize is (x + y)^2 = (x + t x)^2 = x² (1 + t)^2.

Substituting x² from above:

x² = [1 / (t(2 + t))]^{1/2} = 1 / sqrt(t(2 + t)).

Therefore, the expression becomes [1 / sqrt(t(2 + t))] * (1 + t)^2.

So, the problem reduces to minimizing f(t) = (1 + t)^2 / sqrt(t(2 + t)) for t > 0.

Hmm, okay. Let me write that as f(t) = (1 + t)^2 / [sqrt(t(t + 2))]. Let me square it to make it easier? Maybe not necessary. Let me try to take derivative.

Alternatively, maybe make substitution. Let me set u = t + 1 or something else. Alternatively, let me write f(t) as [(1 + t)^2] / [sqrt(t(t + 2))]. Let me square f(t) to make it easier for differentiation:

[f(t)]² = (1 + t)^4 / [t(t + 2)].

Let me denote g(t) = (1 + t)^4 / [t(t + 2)]. Then, minimizing f(t) is equivalent to minimizing g(t).

So, let me compute derivative of g(t). Let me compute g'(t):

First, write g(t) = (1 + t)^4 / [t(t + 2)] = (1 + t)^4 / (t² + 2t).

Let me use quotient rule. Let me denote numerator as N = (1 + t)^4, denominator as D = t² + 2t.

Then, g'(t) = [N' D - N D'] / D².

Compute N' = 4(1 + t)^3.

Compute D' = 2t + 2.

Therefore,

g'(t) = [4(1 + t)^3 (t² + 2t) - (1 + t)^4 (2t + 2)] / (t² + 2t)^2.

Factor out common terms in numerator:

Let me factor out (1 + t)^3 (2t + 2) ?

Wait, let me compute numerator:

First term: 4(1 + t)^3 (t² + 2t)

Second term: - (1 + t)^4 (2t + 2)

Let me factor out (1 + t)^3 (2t + 2):

Wait, note that t² + 2t = t(t + 2), and 2t + 2 = 2(t + 1). Let me see:

Let me write numerator:

4(1 + t)^3 * t(t + 2) - (1 + t)^4 * 2(t + 1)

Factor out (1 + t)^3 * 2(t + 1):

Wait, let me see:

First term: 4(1 + t)^3 * t(t + 2)

Second term: - (1 + t)^4 * 2(t + 1) = -2(1 + t)^4 (t + 1)

Wait, maybe factor out 2(1 + t)^3:

Numerator = 2(1 + t)^3 [ 2t(t + 2) - (1 + t)(t + 1) ]

Let me compute the bracket:

First part: 2t(t + 2) = 2t² + 4t

Second part: (1 + t)(t + 1) = (t + 1)^2 = t² + 2t + 1

Therefore, the bracket becomes:

2t² + 4t - (t² + 2t + 1) = 2t² + 4t - t² - 2t - 1 = t² + 2t - 1

Therefore, numerator is 2(1 + t)^3 (t² + 2t - 1)

Therefore, g'(t) = [2(1 + t)^3 (t² + 2t - 1)] / (t² + 2t)^2

Set derivative equal to zero:

Numerator must be zero. Since denominator is always positive for t > 0, we have:

2(1 + t)^3 (t² + 2t - 1) = 0

Since t > 0, (1 + t)^3 is always positive, so the critical points occur when t² + 2t - 1 = 0.

Solving t² + 2t - 1 = 0:

t = [-2 ± sqrt(4 + 4)] / 2 = [-2 ± sqrt(8)] / 2 = [-2 ± 2*sqrt(2)] / 2 = -1 ± sqrt(2)

Since t > 0, we take the positive root: t = -1 + sqrt(2) ≈ -1 + 1.414 ≈ 0.414, which is positive. So t = sqrt(2) - 1.

Therefore, critical point at t = sqrt(2) - 1. Let me check if this is a minimum.

We can do second derivative test or check sign changes. Let me check the sign of g'(t) around t = sqrt(2) - 1.

For t slightly less than sqrt(2) - 1, say t approaching from the left, t² + 2t - 1 is negative (since at t = 0, t² + 2t - 1 = -1 < 0, and the quadratic crosses zero at t = sqrt(2)-1). Therefore, for t < sqrt(2)-1, numerator is negative, so g'(t) < 0. For t > sqrt(2)-1, numerator is positive, so g'(t) > 0. Therefore, the function g(t) has a minimum at t = sqrt(2) - 1.

Therefore, the minimum of g(t) occurs at t = sqrt(2) - 1, so the minimum of f(t) is sqrt(g(t)).

Wait, but actually, since we squared f(t) to get g(t), so the minimum of f(t) corresponds to the minimum of g(t). Therefore, once we find the minimum of g(t), we can take square root to get the minimum of f(t).

So let me compute g(t) at t = sqrt(2) - 1.

First, compute t = sqrt(2) - 1. Let me compute t + 1 = sqrt(2). Also, t + 2 = sqrt(2) - 1 + 2 = sqrt(2) + 1.

So, let me compute (1 + t)^4 / [t(t + 2)].

First, (1 + t)^4 = (sqrt(2))^4 = (2)^2 = 4.

Then, t(t + 2) = (sqrt(2) - 1)(sqrt(2) + 1) = (sqrt(2))^2 - 1^2 = 2 - 1 = 1.

Therefore, g(t) = 4 / 1 = 4. Therefore, [f(t)]² = 4, so f(t) = 2. Therefore, the minimum value of (x + y)^2 is 2, so the minimum of (x + y)(y + z) is 2? Wait, but we assumed x = z. So if this is the case, then the minimum value is 2? But wait, let me check.

Wait, if (x + y)^2 = 2, then (x + y)(y + z) = (x + y)^2 = 2. But is this the actual minimum? Wait, but we assumed x = z. So maybe there is a lower value if we don't assume x = z? Or maybe not? Let me check.

Alternatively, maybe this is indeed the minimum. Let me check with actual numbers. Let me compute x, y, z.

Given t = sqrt(2) - 1, which is y/x. So y = t x. Also, from earlier, x^4 = 1 / [t(2 + t)].

Let me compute t(2 + t):

t(2 + t) = (sqrt(2) - 1)(2 + sqrt(2) - 1) = (sqrt(2) - 1)(sqrt(2) + 1) = 2 - 1 = 1. Therefore, x^4 = 1 / 1 = 1, so x = 1. Therefore, x = 1, z = x = 1, and y = t x = (sqrt(2) - 1)*1 = sqrt(2) - 1.

Therefore, x = 1, y = sqrt(2) - 1, z = 1.

Let me check the constraint: xyz(x + y + z) = 1 * (sqrt(2) - 1) * 1 * (1 + sqrt(2) - 1 + 1) = (sqrt(2) - 1) * (sqrt(2) + 1) = 2 - 1 = 1. Correct. So the constraint is satisfied.

Then, compute (x + y)(y + z) = (1 + sqrt(2) - 1)(sqrt(2) - 1 + 1) = (sqrt(2))(sqrt(2)) = 2. So indeed, the value is 2. So under the assumption x = z, we get the value 2. But is this the actual minimum?

Wait, but maybe there's a lower value if we don't assume x = z. How can I be sure?

Alternatively, maybe I can try another approach without assuming x = z. Let me think.

Let me consider variables x, y, z. Let me denote S = x + y + z. The constraint is xyzS = 1. We need to minimize (x + y)(y + z).

Note that (x + y)(y + z) = y(x + y + z) + xz = yS + xz. Therefore, (x + y)(y + z) = yS + xz.

So, our target expression is yS + xz. Given that xyzS = 1. So, maybe we can express variables in terms of S or something else.

Alternatively, since we have xyzS = 1, maybe express xz = 1/(yS). Therefore, substituting into the expression:

(x + y)(y + z) = yS + xz = yS + 1/(yS). Therefore, the expression becomes yS + 1/(yS). Let me denote u = yS. Then, the expression is u + 1/u. Which is minimized when u = 1, giving minimum value 2. Wait, that seems too easy. Wait, but is u = yS arbitrary? Or is there a constraint on u?

Wait, if we can set u = yS, then the expression is u + 1/u, which has minimum 2. But is u arbitrary? Let me check.

Wait, but u = yS. However, S = x + y + z. So, if we can have u = yS being any positive real number, then the minimum would be 2. However, there might be constraints on u based on the relationship between variables. Therefore, we need to check if u can indeed be 1.

So, if u = 1, then yS = 1. Also, from the constraint xyzS = 1. Since yS = 1, then xyzS = x * 1 = 1 => x = 1. Therefore, if x = 1, and yS = 1, then we need to check if such variables exist.

But S = x + y + z = 1 + y + z. Also, yS = y(1 + y + z) = 1. So, we have:

y(1 + y + z) = 1. Also, we need to find z such that this holds. However, there are infinitely many solutions here. For example, if we set z = something. Wait, but we also have the original expression (x + y)(y + z) = yS + xz = 1 + xz. Wait, but if u = yS = 1, then the expression is 1 + xz. But earlier, I thought that (x + y)(y + z) = yS + xz. Wait, let me check again:

Earlier, I said (x + y)(y + z) = yS + xz. Let me verify:

(x + y)(y + z) = xy + xz + y² + yz = y(x + y + z) + xz = yS + xz. Yes, that's correct. Therefore, if yS = 1, then (x + y)(y + z) = 1 + xz. But from the constraint xyzS = 1, since yS = 1, then x * 1 = 1 => x = 1. Therefore, (x + y)(y + z) = 1 + xz = 1 + z. But z is a variable here. Wait, but how do we know z?

Wait, if x = 1, and yS = 1, then S = 1 + y + z. Therefore, y(1 + y + z) = 1. Let me denote that equation:

y(1 + y + z) = 1. Let me solve for z:

z = (1 / y) - 1 - y.

Therefore, z = (1 - y - y²)/y.

But z must be positive. Therefore, 1 - y - y² > 0 => y² + y - 1 < 0. Solving y² + y - 1 = 0, the roots are y = [-1 ± sqrt(1 + 4)] / 2 = [-1 ± sqrt(5)] / 2. The positive root is (-1 + sqrt(5))/2 ≈ (-1 + 2.236)/2 ≈ 0.618. Therefore, y must be less than (-1 + sqrt(5))/2 ≈ 0.618 for z to be positive.

Therefore, if we take y such that y < 0.618, then z is positive. However, in this case, (x + y)(y + z) = 1 + z. But z = (1 - y - y²)/y. Therefore, substituting:

(x + y)(y + z) = 1 + (1 - y - y²)/y = 1 + 1/y - 1 - y = 1/y - y.

Therefore, the expression becomes 1/y - y. So, we need to minimize 1/y - y, given that y < (-1 + sqrt(5))/2 ≈ 0.618. Wait, but if we can choose y such that 1/y - y is minimized?

Wait, but earlier I thought that if u = yS = 1, then the expression is u + 1/u = 2. But that seems conflicting with this result. Where is the mistake here?

Wait, earlier I thought that (x + y)(y + z) = yS + xz, and since xyzS = 1, then xz = 1/(yS). Therefore, if yS = 1, then xz = 1/(yS) = 1. Therefore, (x + y)(y + z) = 1 + 1 = 2. But according to the other calculation, it's 1/y - y. Which is conflicting. Therefore, there must be a mistake in my reasoning.

Wait, let me check again. If yS = 1, then xz = 1/(yS) = 1. Therefore, (x + y)(y + z) = yS + xz = 1 + 1 = 2. However, when I substituted variables, I found that (x + y)(y + z) = 1 + z. But according to this, it's 2. Therefore, there must be an error in substitution.

Wait, let me re-examine:

If x = 1, and yS = 1, then S = 1 + y + z. Therefore, y(1 + y + z) = 1. Therefore, yS = 1. Then, (x + y)(y + z) = yS + xz = 1 + xz. But since xyzS = 1, and yS = 1, then x * 1 = 1 => x = 1. Therefore, xz = z. Therefore, (x + y)(y + z) = 1 + z. But also, from yS = 1, we have z = (1 - y - y²)/y. Therefore, substituting into 1 + z:

1 + z = 1 + (1 - y - y²)/y = 1 + 1/y - 1 - y = 1/y - y. Therefore, according to this, (x + y)(y + z) = 1/y - y. But according to the previous reasoning, it should be 2. Therefore, there is a contradiction here. Which means that my assumption that (x + y)(y + z) = yS + xz and then xz = 1/(yS) is wrong?

Wait, let me check again:

Original expression: (x + y)(y + z) = yS + xz. Correct.

Constraint: xyzS = 1. Therefore, xz = 1/(yS). Therefore, substituting into the expression: yS + 1/(yS). Therefore, if yS = 1, then expression is 1 + 1 = 2. However, when I substituted variables, I found that it's 1/y - y. Therefore, there must be a mistake in substitution. Let me check again:

If x = 1, and yS = 1, then S = 1 + y + z. Therefore, y(1 + y + z) = 1. Therefore, yS = 1. Then, xz = 1/(yS) = 1. Therefore, xz = 1. Since x = 1, then z = 1. Therefore, z = 1. Therefore, substituting back into S = 1 + y + z = 1 + y + 1 = 2 + y. Therefore, yS = y(2 + y) = 1. Therefore, y² + 2y - 1 = 0. Solving this quadratic equation: y = [-2 ± sqrt(4 + 4)] / 2 = [-2 ± sqrt(8)] / 2 = [-2 ± 2*sqrt(2)] / 2 = -1 ± sqrt(2). Since y > 0, y = -1 + sqrt(2) ≈ 0.414. Therefore, z = 1. Therefore, (x + y)(y + z) = (1 + y)(y + 1) = (1 + y)^2. Which is (1 + (-1 + sqrt(2)))^2 = (sqrt(2))^2 = 2. Which matches the previous result. Therefore, my mistake earlier was in the substitution where I thought z = (1 - y - y²)/y. Wait, let me check that again.

Earlier, I said:

From y(1 + y + z) = 1, solving for z:

z = (1 / y) - 1 - y. But if x = 1, and S = 1 + y + z, then yS = y(1 + y + z) = 1. Therefore, z = (1 / y) - 1 - y. However, if we also have xz = 1, since x = 1, then z = 1. Therefore, substituting z = 1 into z = (1 / y) - 1 - y:

1 = (1 / y) - 1 - y => 1 + 1 + y = 1 / y => 2 + y = 1 / y => y(2 + y) = 1 => y² + 2y - 1 = 0, which is the same equation as before. Therefore, my previous mistake was that I thought (x + y)(y + z) = 1 + z, but actually, since xz = 1, then (x + y)(y + z) = yS + xz = 1 + 1 = 2. Therefore, the previous calculation where I thought it was 1/y - y was wrong because I incorrectly substituted z. Therefore, the correct value is indeed 2. Therefore, if we can have yS = 1, then the expression is 2, which is the minimum of u + 1/u. Therefore, this suggests that the minimum value is 2. However, this seems to contradict my previous thought where I thought that maybe there is a lower value. But according to this, if we can set u = yS = 1, then the expression is minimized at 2. However, is u = yS allowed to be 1? Yes, as shown in the example with x = 1, y = sqrt(2) - 1, z = 1, which satisfies the constraint and gives the expression value 2.

But then why did I get confused earlier? Because I thought that (x + y)(y + z) = 1 + z, but that was incorrect. Actually, since xz = 1, then (x + y)(y + z) = yS + xz = 1 + 1 = 2. Therefore, this seems to confirm that the minimum is 2. However, let me check if there are other possibilities where u + 1/u can be less than 2. But since u + 1/u ≥ 2 by AM-GM, equality holds when u = 1. Therefore, the minimum value is indeed 2. Therefore, regardless of the variables, the minimum value of (x + y)(y + z) is 2. Therefore, the answer is 2. But wait, but in the first approach, when I assumed x = z, I also got 2. Therefore, both approaches lead to the same result. Therefore, maybe 2 is indeed the minimum.

But let me check with another example. Suppose I take different values. Let me try x = 2, y = 1, z = ?

Compute xyz(x + y + z) = 2 * 1 * z * (2 + 1 + z) = 2z(3 + z) = 1. Therefore, 2z(3 + z) = 1 => 2z² + 6z - 1 = 0. Solving quadratic equation: z = [-6 ± sqrt(36 + 8)] / 4 = [-6 ± sqrt(44)] / 4 = [-6 ± 2*sqrt(11)] / 4 = [-3 ± sqrt(11)] / 2. Taking positive root: z = [-3 + sqrt(11)] / 2 ≈ (-3 + 3.316)/2 ≈ 0.316 / 2 ≈ 0.158. Then compute (x + y)(y + z) = (2 + 1)(1 + 0.158) ≈ 3 * 1.158 ≈ 3.474, which is greater than 2. So, higher than 2. Therefore, seems like 2 is indeed the minimum.

Another test case: take x approaching 0. Suppose x approaches 0, then to satisfy xyz(x + y + z) = 1, since x approaches 0, we need y or z to approach infinity. Let me see. Suppose x approaches 0, and let me set y = z. Then, the constraint becomes x * y * y * (x + 2y) = x y² (x + 2y) = 1. If x approaches 0, then x y² * 2y = 2x y³ ≈ 1. Therefore, x ≈ 1/(2 y³). Then, compute (x + y)(y + z) = (x + y)(y + y) = (x + y)(2y) ≈ (0 + y)(2y) = 2 y². Since x ≈ 1/(2 y³), then as x approaches 0, y can be any positive number. However, if we want to minimize 2 y², we need y approaching 0, but then x ≈ 1/(2 y³) approaches infinity. However, in this case, the expression (x + y)(y + z) approaches 0? Wait, no. Wait, if y approaches 0, then 2 y² approaches 0. But how does that reconcile with the constraint? Wait, if y approaches 0, then x ≈ 1/(2 y³) approaches infinity. However, the expression (x + y)(y + z) = (x + y)(2y) ≈ x * 2y. Since x ≈ 1/(2 y³), then x * 2y ≈ (1/(2 y³)) * 2y = 1 / y², which approaches infinity as y approaches 0. Therefore, in this case, the expression approaches infinity. Therefore, not helpful for minimum.

Alternatively, if I take y approaching infinity, then x ≈ 1/(2 y³) approaches 0, and (x + y)(2y) ≈ y * 2y = 2 y² approaches infinity. Therefore, in this case, the expression also approaches infinity. Therefore, no minimum here. Therefore, seems like 2 is indeed the minimum.

Alternatively, let me try another case where variables are not symmetric. Let me take x = 1, y = 1, then find z such that xyz(x + y + z) = 1. So, 1 * 1 * z * (1 + 1 + z) = z(2 + z) = 1. Therefore, z² + 2z - 1 = 0. Solutions z = [-2 ± sqrt(4 + 4)] / 2 = [-2 ± sqrt(8)] / 2 = -1 ± sqrt(2). Taking positive root, z = -1 + sqrt(2) ≈ 0.414. Then compute (x + y)(y + z) = (1 + 1)(1 + 0.414) ≈ 2 * 1.414 ≈ 2.828, which is greater than 2. Therefore, again, higher than 2.

Therefore, all these examples suggest that 2 is indeed the minimum. Therefore, maybe the answer is 2. However, let me check if there is a way to formally prove that the minimum is 2.

Earlier, I considered that (x + y)(y + z) = yS + xz, and since xyzS = 1, then xz = 1/(yS). Therefore, (x + y)(y + z) = yS + 1/(yS). Let me denote u = yS. Then, the expression becomes u + 1/u, which by AM-GM is ≥ 2, with equality when u = 1. Therefore, the minimum value is 2, achieved when u = 1, i.e., yS = 1. Therefore, this is a general proof, without assuming x = z. Therefore, regardless of the values of x, y, z, as long as they satisfy the constraint, the expression (x + y)(y + z) is ≥ 2, with equality when yS = 1 and xz = 1. Therefore, the minimum value is 2.

But wait, how do we know that when yS = 1 and xz = 1, there exist positive real numbers x, y, z satisfying the constraint? Well, we saw in the example with x = 1, y = sqrt(2) - 1, z = 1, which satisfies xyz(x + y + z) = 1 * (sqrt(2) - 1) * 1 * (1 + sqrt(2) - 1 + 1) = (sqrt(2) - 1)(sqrt(2) + 1) = 1. Therefore, such numbers exist. Therefore, the minimum is indeed achievable, so the minimum value is 2.

Therefore, the answer is \boxed{2}.

**Final Answer**
\boxed{2}

To find the **minimum value** of the expression
$$
(x + y)(y + z)
$$
given the constraint
$$
xyz(x + y + z) = 1,
$$
where $ x, y, z $ are **positive real numbers**, we proceed with a combination of algebraic manipulation and inequality techniques.

---

### Step 1: Expand the Expression

We begin by expanding the expression:
$$
(x + y)(y + z) = xy + xz + y^2 + yz.
$$
This can be rewritten as:
$$
(x + y)(y + z) = y(x + y + z) + xz.
$$
Let $ S = x + y + z $, then the expression becomes:
$$
(x + y)(y + z) = yS + xz.
$$

---

### Step 2: Use the Constraint

From the given constraint:
$$
xyzS = 1 \Rightarrow xz = \frac{1}{yS}.
$$
Substituting this into the expression:
$$
(x + y)(y + z) = yS + \frac{1}{yS}.
$$

Let $ u = yS $. Then the expression becomes:
$$
(x + y)(y + z) = u + \frac{1}{u}.
$$

---

### Step 3: Apply AM-GM Inequality

We now minimize the expression $ u + \frac{1}{u} $ for $ u > 0 $. By the **AM-GM inequality**:
$$
u + \frac{1}{u} \geq 2,
$$
with equality if and only if $ u = 1 $.

Thus, the **minimum value** of $ (x + y)(y + z) $ is:
$$
\boxed{2}.
$$

---

### Step 4: Verify Achievability of the Minimum

We must confirm that the minimum value of 2 is **achievable** under the given constraint. That is, we need to find positive real numbers $ x, y, z $ such that:
- $ xyz(x + y + z) = 1 $,
- and $ (x + y)(y + z) = 2 $.

Let’s choose:
- $ x = 1 $,
- $ z = 1 $,
- $ y = \sqrt{2} - 1 $.

Then:
- $ x + y + z = 1 + (\sqrt{2} - 1) + 1 = \sqrt{2} $,
- $ xyz(x + y + z) = 1 \cdot (\sqrt{2} - 1) \cdot 1 \cdot \sqrt{2} = (\sqrt{2} - 1)\sqrt{2} = 2 - 1 = 1 $, which satisfies the constraint.

Now compute:
$$
(x + y)(y + z) = (1 + \sqrt{2} - 1)(\sqrt{2} - 1 + 1) = \sqrt{2} \cdot \sqrt{2} = 2.
$$

Thus, the minimum value is **attainable**.

---

### Final Answer

$$
\boxed{2}
$$

[Assistant]
Explanation:
Okay, so I need to find the minimum value of (x + y)(y + z) given that x, y, z are positive real numbers and satisfy the condition xyz(x + y + z) = 1. Hmm, let me think about how to approach this.

First, I remember that optimization problems with constraints often use methods like Lagrange multipliers, but since this is likely a problem from algebra or inequalities, maybe I can use AM-GM inequality or some substitution. Let me try to recall the AM-GM inequality. It states that for non-negative numbers, the arithmetic mean is greater than or equal to the geometric mean. Maybe I can apply that here.

But before jumping into inequalities, let me write down what I need to minimize and the constraint:

Minimize (x + y)(y + z) subject to xyz(x + y + z) = 1.

Let me expand (x + y)(y + z) to see if that helps. Expanding it:

(x + y)(y + z) = xy + xz + y² + yz.

Hmm, not sure if that helps directly. Maybe I can think of variables substitution. Let me consider variables a, b, c or something else. Alternatively, maybe express variables in terms of each other.

Alternatively, since there are three variables and one constraint, maybe I can reduce the number of variables. Let me think. Let me try to set variables such that the constraint becomes simpler. For example, maybe set x = z? Because the expression (x + y)(y + z) is symmetric in x and z. If I assume x = z, maybe that would simplify things. Let me try that.

Let me assume x = z. Then, the constraint becomes x * y * x * (x + y + x) = x² y (2x + y) = 1. And the expression to minimize becomes (x + y)(y + x) = (x + y)². So, if I can find the minimum of (x + y)² given that x² y (2x + y) = 1. Maybe this is easier?

Alternatively, maybe not. Let me check if assuming x = z is valid. Since the original problem is symmetric in x and z, maybe the minimum occurs when x = z. But I need to verify that. However, since I don't know yet, maybe I should proceed with this assumption and see if I can find a solution, and then check if it's indeed the minimum.

So, assuming x = z, then we have:

Let me denote x = z. Let me set variables as x, y, x. Then, the constraint is x² y (2x + y) = 1. Let me denote t = y/x. Let me set t as a positive real number. Then y = t x. Then substituting into the constraint:

x² * (t x) * (2x + t x) = x² * t x * x (2 + t) = t (2 + t) x^4 = 1. Therefore, x^4 = 1 / [t(2 + t)].

So x = [1 / (t(2 + t))]^{1/4}.

Then, the expression to minimize is (x + y)^2 = (x + t x)^2 = x² (1 + t)^2.

Substituting x² from above:

x² = [1 / (t(2 + t))]^{1/2} = 1 / sqrt(t(2 + t)).

Therefore, the expression becomes [1 / sqrt(t(2 + t))] * (1 + t)^2.

So, the problem reduces to minimizing f(t) = (1 + t)^2 / sqrt(t(2 + t)) for t > 0.

Hmm, okay. Let me write that as f(t) = (1 + t)^2 / [sqrt(t(t + 2))]. Let me square it to make it easier? Maybe not necessary. Let me try to take derivative.

Alternatively, maybe make substitution. Let me set u = t + 1 or something else. Alternatively, let me write f(t) as [(1 + t)^2] / [sqrt(t(t + 2))]. Let me square f(t) to make it easier for differentiation:

[f(t)]² = (1 + t)^4 / [t(t + 2)].

Let me denote g(t) = (1 + t)^4 / [t(t + 2)]. Then, minimizing f(t) is equivalent to minimizing g(t).

So, let me compute derivative of g(t). Let me compute g'(t):

First, write g(t) = (1 + t)^4 / [t(t + 2)] = (1 + t)^4 / (t² + 2t).

Let me use quotient rule. Let me denote numerator as N = (1 + t)^4, denominator as D = t² + 2t.

Then, g'(t) = [N' D - N D'] / D².

Compute N' = 4(1 + t)^3.

Compute D' = 2t + 2.

Therefore,

g'(t) = [4(1 + t)^3 (t² + 2t) - (1 + t)^4 (2t + 2)] / (t² + 2t)^2.

Factor out common terms in numerator:

Let me factor out (1 + t)^3 (2t + 2) ?

Wait, let me compute numerator:

First term: 4(1 + t)^3 (t² + 2t)

Second term: - (1 + t)^4 (2t + 2)

Let me factor out (1 + t)^3 (2t + 2):

Wait, note that t² + 2t = t(t + 2), and 2t + 2 = 2(t + 1). Let me see:

Let me write numerator:

4(1 + t)^3 * t(t + 2) - (1 + t)^4 * 2(t + 1)

Factor out (1 + t)^3 * 2(t + 1):

Wait, let me see:

First term: 4(1 + t)^3 * t(t + 2)

Second term: - (1 + t)^4 * 2(t + 1) = -2(1 + t)^4 (t + 1)

Wait, maybe factor out 2(1 + t)^3:

Numerator = 2(1 + t)^3 [ 2t(t + 2) - (1 + t)(t + 1) ]

Let me compute the bracket:

First part: 2t(t + 2) = 2t² + 4t

Second part: (1 + t)(t + 1) = (t + 1)^2 = t² + 2t + 1

Therefore, the bracket becomes:

2t² + 4t - (t² + 2t + 1) = 2t² + 4t - t² - 2t - 1 = t² + 2t - 1

Therefore, numerator is 2(1 + t)^3 (t² + 2t - 1)

Therefore, g'(t) = [2(1 + t)^3 (t² + 2t - 1)] / (t² + 2t)^2

Set derivative equal to zero:

Numerator must be zero. Since denominator is always positive for t > 0, we have:

2(1 + t)^3 (t² + 2t - 1) = 0

Since t > 0, (1 + t)^3 is always positive, so the critical points occur when t² + 2t - 1 = 0.

Solving t² + 2t - 1 = 0:

t = [-2 ± sqrt(4 + 4)] / 2 = [-2 ± sqrt(8)] / 2 = [-2 ± 2*sqrt(2)] / 2 = -1 ± sqrt(2)

Since t > 0, we take the positive root: t = -1 + sqrt(2) ≈ -1 + 1.414 ≈ 0.414, which is positive. So t = sqrt(2) - 1.

Therefore, critical point at t = sqrt(2) - 1. Let me check if this is a minimum.

We can do second derivative test or check sign changes. Let me check the sign of g'(t) around t = sqrt(2) - 1.

For t slightly less than sqrt(2) - 1, say t approaching from the left, t² + 2t - 1 is negative (since at t = 0, t² + 2t - 1 = -1 < 0, and the quadratic crosses zero at t = sqrt(2)-1). Therefore, for t < sqrt(2)-1, numerator is negative, so g'(t) < 0. For t > sqrt(2)-1, numerator is positive, so g'(t) > 0. Therefore, the function g(t) has a minimum at t = sqrt(2) - 1.

Therefore, the minimum of g(t) occurs at t = sqrt(2) - 1, so the minimum of f(t) is sqrt(g(t)).

Wait, but actually, since we squared f(t) to get g(t), so the minimum of f(t) corresponds to the minimum of g(t). Therefore, once we find the minimum of g(t), we can take square root to get the minimum of f(t).

So let me compute g(t) at t = sqrt(2) - 1.

First, compute t = sqrt(2) - 1. Let me compute t + 1 = sqrt(2). Also, t + 2 = sqrt(2) - 1 + 2 = sqrt(2) + 1.

So, let me compute (1 + t)^4 / [t(t + 2)].

First, (1 + t)^4 = (sqrt(2))^4 = (2)^2 = 4.

Then, t(t + 2) = (sqrt(2) - 1)(sqrt(2) + 1) = (sqrt(2))^2 - 1^2 = 2 - 1 = 1.

Therefore, g(t) = 4 / 1 = 4. Therefore, [f(t)]² = 4, so f(t) = 2. Therefore, the minimum value of (x + y)^2 is 2, so the minimum of (x + y)(y + z) is 2? Wait, but we assumed x = z. So if this is the case, then the minimum value is 2? But wait, let me check.

Wait, if (x + y)^2 = 2, then (x + y)(y + z) = (x + y)^2 = 2. But is this the actual minimum? Wait, but we assumed x = z. So maybe there is a lower value if we don't assume x = z? Or maybe not? Let me check.

Alternatively, maybe this is indeed the minimum. Let me check with actual numbers. Let me compute x, y, z.

Given t = sqrt(2) - 1, which is y/x. So y = t x. Also, from earlier, x^4 = 1 / [t(2 + t)].

Let me compute t(2 + t):

t(2 + t) = (sqrt(2) - 1)(2 + sqrt(2) - 1) = (sqrt(2) - 1)(sqrt(2) + 1) = 2 - 1 = 1. Therefore, x^4 = 1 / 1 = 1, so x = 1. Therefore, x = 1, z = x = 1, and y = t x = (sqrt(2) - 1)*1 = sqrt(2) - 1.

Therefore, x = 1, y = sqrt(2) - 1, z = 1.

Let me check the constraint: xyz(x + y + z) = 1 * (sqrt(2) - 1) * 1 * (1 + sqrt(2) - 1 + 1) = (sqrt(2) - 1) * (sqrt(2) + 1) = 2 - 1 = 1. Correct. So the constraint is satisfied.

Then, compute (x + y)(y + z) = (1 + sqrt(2) - 1)(sqrt(2) - 1 + 1) = (sqrt(2))(sqrt(2)) = 2. So indeed, the value is 2. So under the assumption x = z, we get the value 2. But is this the actual minimum?

Wait, but maybe there's a lower value if we don't assume x = z. How can I be sure?

Alternatively, maybe I can try another approach without assuming x = z. Let me think.

Let me consider variables x, y, z. Let me denote S = x + y + z. The constraint is xyzS = 1. We need to minimize (x + y)(y + z).

Note that (x + y)(y + z) = y(x + y + z) + xz = yS + xz. Therefore, (x + y)(y + z) = yS + xz.

So, our target expression is yS + xz. Given that xyzS = 1. So, maybe we can express variables in terms of S or something else.

Alternatively, since we have xyzS = 1, maybe express xz = 1/(yS). Therefore, substituting into the expression:

(x + y)(y + z) = yS + xz = yS + 1/(yS). Therefore, the expression becomes yS + 1/(yS). Let me denote u = yS. Then, the expression is u + 1/u. Which is minimized when u = 1, giving minimum value 2. Wait, that seems too easy. Wait, but is u = yS arbitrary? Or is there a constraint on u?

Wait, if we can set u = yS, then the expression is u + 1/u, which has minimum 2. But is u arbitrary? Let me check.

Wait, but u = yS. However, S = x + y + z. So, if we can have u = yS being any positive real number, then the minimum would be 2. However, there might be constraints on u based on the relationship between variables. Therefore, we need to check if u can indeed be 1.

So, if u = 1, then yS = 1. Also, from the constraint xyzS = 1. Since yS = 1, then xyzS = x * 1 = 1 => x = 1. Therefore, if x = 1, and yS = 1, then we need to check if such variables exist.

But S = x + y + z = 1 + y + z. Also, yS = y(1 + y + z) = 1. So, we have:

y(1 + y + z) = 1. Also, we need to find z such that this holds. However, there are infinitely many solutions here. For example, if we set z = something. Wait, but we also have the original expression (x + y)(y + z) = yS + xz = 1 + xz. Wait, but if u = yS = 1, then the expression is 1 + xz. But earlier, I thought that (x + y)(y + z) = yS + xz. Wait, let me check again:

Earlier, I said (x + y)(y + z) = yS + xz. Let me verify:

(x + y)(y + z) = xy + xz + y² + yz = y(x + y + z) + xz = yS + xz. Yes, that's correct. Therefore, if yS = 1, then (x + y)(y + z) = 1 + xz. But from the constraint xyzS = 1, since yS = 1, then x * 1 = 1 => x = 1. Therefore, (x + y)(y + z) = 1 + xz = 1 + z. But z is a variable here. Wait, but how do we know z?

Wait, if x = 1, and yS = 1, then S = 1 + y + z. Therefore, y(1 + y + z) = 1. Let me denote that equation:

y(1 + y + z) = 1. Let me solve for z:

z = (1 / y) - 1 - y.

Therefore, z = (1 - y - y²)/y.

But z must be positive. Therefore, 1 - y - y² > 0 => y² + y - 1 < 0. Solving y² + y - 1 = 0, the roots are y = [-1 ± sqrt(1 + 4)] / 2 = [-1 ± sqrt(5)] / 2. The positive root is (-1 + sqrt(5))/2 ≈ (-1 + 2.236)/2 ≈ 0.618. Therefore, y must be less than (-1 + sqrt(5))/2 ≈ 0.618 for z to be positive.

Therefore, if we take y such that y < 0.618, then z is positive. However, in this case, (x + y)(y + z) = 1 + z. But z = (1 - y - y²)/y. Therefore, substituting:

(x + y)(y + z) = 1 + (1 - y - y²)/y = 1 + 1/y - 1 - y = 1/y - y.

Therefore, the expression becomes 1/y - y. So, we need to minimize 1/y - y, given that y < (-1 + sqrt(5))/2 ≈ 0.618. Wait, but if we can choose y such that 1/y - y is minimized?

Wait, but earlier I thought that if u = yS = 1, then the expression is u + 1/u = 2. But that seems conflicting with this result. Where is the mistake here?

Wait, earlier I thought that (x + y)(y + z) = yS + xz, and since xyzS = 1, then xz = 1/(yS). Therefore, if yS = 1, then xz = 1/(yS) = 1. Therefore, (x + y)(y + z) = 1 + 1 = 2. But according to the other calculation, it's 1/y - y. Which is conflicting. Therefore, there must be a mistake in my reasoning.

Wait, let me check again. If yS = 1, then xz = 1/(yS) = 1. Therefore, (x + y)(y + z) = yS + xz = 1 + 1 = 2. However, when I substituted variables, I found that (x + y)(y + z) = 1 + z. But according to this, it's 2. Therefore, there must be an error in substitution.

Wait, let me re-examine:

If x = 1, and yS = 1, then S = 1 + y + z. Therefore, y(1 + y + z) = 1. Therefore, yS = 1. Then, (x + y)(y + z) = yS + xz = 1 + xz. But since xyzS = 1, and yS = 1, then x * 1 = 1 => x = 1. Therefore, xz = z. Therefore, (x + y)(y + z) = 1 + z. But also, from yS = 1, we have z = (1 - y - y²)/y. Therefore, substituting into 1 + z:

1 + z = 1 + (1 - y - y²)/y = 1 + 1/y - 1 - y = 1/y - y. Therefore, according to this, (x + y)(y + z) = 1/y - y. But according to the previous reasoning, it should be 2. Therefore, there is a contradiction here. Which means that my assumption that (x + y)(y + z) = yS + xz and then xz = 1/(yS) is wrong?

Wait, let me check again:

Original expression: (x + y)(y + z) = yS + xz. Correct.

Constraint: xyzS = 1. Therefore, xz = 1/(yS). Therefore, substituting into the expression: yS + 1/(yS). Therefore, if yS = 1, then expression is 1 + 1 = 2. However, when I substituted variables, I found that it's 1/y - y. Therefore, there must be a mistake in substitution. Let me check again:

If x = 1, and yS = 1, then S = 1 + y + z. Therefore, y(1 + y + z) = 1. Therefore, yS = 1. Then, xz = 1/(yS) = 1. Therefore, xz = 1. Since x = 1, then z = 1. Therefore, z = 1. Therefore, substituting back into S = 1 + y + z = 1 + y + 1 = 2 + y. Therefore, yS = y(2 + y) = 1. Therefore, y² + 2y - 1 = 0. Solving this quadratic equation: y = [-2 ± sqrt(4 + 4)] / 2 = [-2 ± sqrt(8)] / 2 = [-2 ± 2*sqrt(2)] / 2 = -1 ± sqrt(2). Since y > 0, y = -1 + sqrt(2) ≈ 0.414. Therefore, z = 1. Therefore, (x + y)(y + z) = (1 + y)(y + 1) = (1 + y)^2. Which is (1 + (-1 + sqrt(2)))^2 = (sqrt(2))^2 = 2. Which matches the previous result. Therefore, my mistake earlier was in the substitution where I thought z = (1 - y - y²)/y. Wait, let me check that again.

Earlier, I said:

From y(1 + y + z) = 1, solving for z:

z = (1 / y) - 1 - y. But if x = 1, and S = 1 + y + z, then yS = y(1 + y + z) = 1. Therefore, z = (1 / y) - 1 - y. However, if we also have xz = 1, since x = 1, then z = 1. Therefore, substituting z = 1 into z = (1 / y) - 1 - y:

1 = (1 / y) - 1 - y => 1 + 1 + y = 1 / y => 2 + y = 1 / y => y(2 + y) = 1 => y² + 2y - 1 = 0, which is the same equation as before. Therefore, my previous mistake was that I thought (x + y)(y + z) = 1 + z, but actually, since xz = 1, then (x + y)(y + z) = yS + xz = 1 + 1 = 2. Therefore, the previous calculation where I thought it was 1/y - y was wrong because I incorrectly substituted z. Therefore, the correct value is indeed 2. Therefore, if we can have yS = 1, then the expression is 2, which is the minimum of u + 1/u. Therefore, this suggests that the minimum value is 2. However, this seems to contradict my previous thought where I thought that maybe there is a lower value. But according to this, if we can set u = yS = 1, then the expression is minimized at 2. However, is u = yS allowed to be 1? Yes, as shown in the example with x = 1, y = sqrt(2) - 1, z = 1, which satisfies the constraint and gives the expression value 2.

But then why did I get confused earlier? Because I thought that (x + y)(y + z) = 1 + z, but that was incorrect. Actually, since xz = 1, then (x + y)(y + z) = yS + xz = 1 + 1 = 2. Therefore, this seems to confirm that the minimum is 2. However, let me check if there are other possibilities where u + 1/u can be less than 2. But since u + 1/u ≥ 2 by AM-GM, equality holds when u = 1. Therefore, the minimum value is indeed 2. Therefore, regardless of the variables, the minimum value of (x + y)(y + z) is 2. Therefore, the answer is 2. But wait, but in the first approach, when I assumed x = z, I also got 2. Therefore, both approaches lead to the same result. Therefore, maybe 2 is indeed the minimum.

But let me check with another example. Suppose I take different values. Let me try x = 2, y = 1, z = ?

Compute xyz(x + y + z) = 2 * 1 * z * (2 + 1 + z) = 2z(3 + z) = 1. Therefore, 2z(3 + z) = 1 => 2z² + 6z - 1 = 0. Solving quadratic equation: z = [-6 ± sqrt(36 + 8)] / 4 = [-6 ± sqrt(44)] / 4 = [-6 ± 2*sqrt(11)] / 4 = [-3 ± sqrt(11)] / 2. Taking positive root: z = [-3 + sqrt(11)] / 2 ≈ (-3 + 3.316)/2 ≈ 0.316 / 2 ≈ 0.158. Then compute (x + y)(y + z) = (2 + 1)(1 + 0.158) ≈ 3 * 1.158 ≈ 3.474, which is greater than 2. So, higher than 2. Therefore, seems like 2 is indeed the minimum.

Another test case: take x approaching 0. Suppose x approaches 0, then to satisfy xyz(x + y + z) = 1, since x approaches 0, we need y or z to approach infinity. Let me see. Suppose x approaches 0, and let me set y = z. Then, the constraint becomes x * y * y * (x + 2y) = x y² (x + 2y) = 1. If x approaches 0, then x y² * 2y = 2x y³ ≈ 1. Therefore, x ≈ 1/(2 y³). Then, compute (x + y)(y + z) = (x + y)(y + y) = (x + y)(2y) ≈ (0 + y)(2y) = 2 y². Since x ≈ 1/(2 y³), then as x approaches 0, y can be any positive number. However, if we want to minimize 2 y², we need y approaching 0, but then x ≈ 1/(2 y³) approaches infinity. However, in this case, the expression (x + y)(y + z) approaches 0? Wait, no. Wait, if y approaches 0, then 2 y² approaches 0. But how does that reconcile with the constraint? Wait, if y approaches 0, then x ≈ 1/(2 y³) approaches infinity. However, the expression (x + y)(y + z) = (x + y)(2y) ≈ x * 2y. Since x ≈ 1/(2 y³), then x * 2y ≈ (1/(2 y³)) * 2y = 1 / y², which approaches infinity as y approaches 0. Therefore, in this case, the expression approaches infinity. Therefore, not helpful for minimum.

Alternatively, if I take y approaching infinity, then x ≈ 1/(2 y³) approaches 0, and (x + y)(2y) ≈ y * 2y = 2 y² approaches infinity. Therefore, in this case, the expression also approaches infinity. Therefore, no minimum here. Therefore, seems like 2 is indeed the minimum.

Alternatively, let me try another case where variables are not symmetric. Let me take x = 1, y = 1, then find z such that xyz(x + y + z) = 1. So, 1 * 1 * z * (1 + 1 + z) = z(2 + z) = 1. Therefore, z² + 2z - 1 = 0. Solutions z = [-2 ± sqrt(4 + 4)] / 2 = [-2 ± sqrt(8)] / 2 = -1 ± sqrt(2). Taking positive root, z = -1 + sqrt(2) ≈ 0.414. Then compute (x + y)(y + z) = (1 + 1)(1 + 0.414) ≈ 2 * 1.414 ≈ 2.828, which is greater than 2. Therefore, again, higher than 2.

Therefore, all these examples suggest that 2 is indeed the minimum. Therefore, maybe the answer is 2. However, let me check if there is a way to formally prove that the minimum is 2.

Earlier, I considered that (x + y)(y + z) = yS + xz, and since xyzS = 1, then xz = 1/(yS). Therefore, (x + y)(y + z) = yS + 1/(yS). Let me denote u = yS. Then, the expression becomes u + 1/u, which by AM-GM is ≥ 2, with equality when u = 1. Therefore, the minimum value is 2, achieved when u = 1, i.e., yS = 1. Therefore, this is a general proof, without assuming x = z. Therefore, regardless of the values of x, y, z, as long as they satisfy the constraint, the expression (x + y)(y + z) is ≥ 2, with equality when yS = 1 and xz = 1. Therefore, the minimum value is 2.

But wait, how do we know that when yS = 1 and xz = 1, there exist positive real numbers x, y, z satisfying the constraint? Well, we saw in the example with x = 1, y = sqrt(2) - 1, z = 1, which satisfies xyz(x + y + z) = 1 * (sqrt(2) - 1) * 1 * (1 + sqrt(2) - 1 + 1) = (sqrt(2) - 1)(sqrt(2) + 1) = 1. Therefore, such numbers exist. Therefore, the minimum is indeed achievable, so the minimum value is 2.

Therefore, the answer is \boxed{2}.

**Final Answer**
\boxed{2}

To find the **minimum value** of the expression
$$
(x + y)(y + z)
$$
given the constraint
$$
xyz(x + y + z) = 1,
$$
where $ x, y, z $ are **positive real numbers**, we proceed with a combination of algebraic manipulation and inequality techniques.

---

### Step 1: Expand the Expression

We begin by expanding the expression:
$$
(x + y)(y + z) = xy + xz + y^2 + yz.
$$
This can be rewritten as:
$$
(x + y)(y + z) = y(x + y + z) + xz.
$$
Let $ S = x + y + z $, then the expression becomes:
$$
(x + y)(y + z) = yS + xz.
$$

---

### Step 2: Use the Constraint

From the given constraint:
$$
xyzS = 1 \Rightarrow xz = \frac{1}{yS}.
$$
Substituting this into the expression:
$$
(x + y)(y + z) = yS + \frac{1}{yS}.
$$

Let $ u = yS $. Then the expression becomes:
$$
(x + y)(y + z) = u + \frac{1}{u}.
$$

---

### Step 3: Apply AM-GM Inequality

We now minimize the expression $ u + \frac{1}{u} $ for $ u > 0 $. By the **AM-GM inequality**:
$$
u + \frac{1}{u} \geq 2,
$$
with equality if and only if $ u = 1 $.

Thus, the **minimum value** of $ (x + y)(y + z) $ is:
$$
\boxed{2}.
$$

---

### Step 4: Verify Achievability of the Minimum

We must confirm that the minimum value of 2 is **achievable** under the given constraint. That is, we need to find positive real numbers $ x, y, z $ such that:
- $ xyz(x + y + z) = 1 $,
- and $ (x + y)(y + z) = 2 $.

Let’s choose:
- $ x = 1 $,
- $ z = 1 $,
- $ y = \sqrt{2} - 1 $.

Then:
- $ x + y + z = 1 + (\sqrt{2} - 1) + 1 = \sqrt{2} $,
- $ xyz(x + y + z) = 1 \cdot (\sqrt{2} - 1) \cdot 1 \cdot \sqrt{2} = (\sqrt{2} - 1)\sqrt{2} = 2 - 1 = 1 $, which satisfies the constraint.

Now compute:
$$
(x + y)(y + z) = (1 + \sqrt{2} - 1)(\sqrt{2} - 1 + 1) = \sqrt{2} \cdot \sqrt{2} = 2.
$$

Thus, the minimum value is **attainable**.

---

### Final Answer

$$
\boxed{2}
$$

Solution:
We can write $(x + y)(y + z)$ as $xz + y(x + y + z).$  By AM-GM,
\[xz + y(x + y + z) \ge 2 \sqrt{(xz)y(x + y + z)} = 2 \sqrt{xyz(x + y + z)} = 2.\]Equality holds when $xz = y(x + y + z) = 1$ and $xyz(x + y + z) = 1.$  For example, we can take $x = 1,$ $y = \sqrt{2} - 1,$ and $z = 1.$  Hence, the minimum value is $\boxed{2}.$


Explanation:
Okay, so I need to find the value of $ b $ in the continued fraction expansion of $ \sqrt[3]{2} $. The given expression is:

$$
\sqrt[3]{2} = a + \cfrac{1}{b + \cfrac{1}{c + \cfrac{1}{d + \dotsb}}}
$$

where $ a, b, c, d $ are positive integers. My task is to compute $ b $.

First, I remember that continued fractions for irrational numbers are infinite and non-repeating. Since $ \sqrt[3]{2} $ is irrational, its continued fraction will be infinite. However, I need to find the first few terms of this continued fraction, specifically the value of $ b $, which is the second term in the expansion.

Let me recall how continued fractions work. For any real number $ x $, the continued fraction expansion is found by taking the integer part, then taking the reciprocal of the fractional part, and repeating the process. So, if I can compute the continued fraction step by step, I can find $ a $, then $ b $, and so on.

Let me start by finding $ a $. Since $ \sqrt[3]{2} $ is approximately... let me calculate that.

I know that $ 1^3 = 1 $, $ 2^3 = 8 $, so $ \sqrt[3]{2} $ is between 1 and 2. Let me compute it more accurately.

Let me recall that $ \sqrt[3]{1} = 1 $, $ \sqrt[3]{8} = 2 $. Let me approximate $ \sqrt[3]{2} $.

I know that $ 1.2^3 = 1.728 $, which is less than 2. $ 1.26^3 = (1.2 + 0.06)^3 $. Let me compute that:

First, $ 1.2^3 = 1.728 $, then $ 1.26^3 $:

Let me compute $ 1.26 \times 1.26 = 1.5876 $, then multiply by 1.26:

$ 1.5876 \times 1.26 $. Let me compute that:

First, 1.5876 * 1 = 1.5876

1.5876 * 0.2 = 0.31752

1.5876 * 0.06 = 0.095256

Adding them up: 1.5876 + 0.31752 = 1.90512 + 0.095256 ≈ 2.000376.

Wow, so $ 1.26^3 ≈ 2.000376 $, which is very close to 2. Therefore, $ \sqrt[3]{2} $ is approximately 1.26. Therefore, the integer part $ a $ is 1.

So, $ a = 1 $. Therefore, we can write:

$$
\sqrt[3]{2} = 1 + \cfrac{1}{b + \cfrac{1}{c + \cfrac{1}{d + \dotsb}}}
$$

So, to find $ b $, I need to compute the continued fraction of $ \sqrt[3]{2} - 1 $, which is approximately 0.26. Let me denote $ x_1 = \sqrt[3]{2} - 1 \approx 0.26 $.

The continued fraction process is as follows: take the reciprocal of $ x_1 $, take the integer part as the next term, and repeat.

So, let me compute $ x_1 = \sqrt[3]{2} - 1 \approx 0.259921049894873... $.

So, $ x_1 \approx 0.259921 $.

Then, the reciprocal of $ x_1 $ is $ 1 / x_1 \approx 1 / 0.259921 ≈ 3.847... $. Therefore, the integer part is 3, so $ b = 3 $. Wait, but let me check this more carefully.

But before I jump to conclusions, maybe I need to do this more accurately. Because if I approximate $ \sqrt[3]{2} $ as 1.26, then $ x_1 = 0.26 $, reciprocal is 3.846... So, the integer part is 3, so $ b = 3 $. But maybe the exact value of $ x_1 $ is slightly different, so maybe the reciprocal is slightly different? Let me check with more accurate value of $ \sqrt[3]{2} $.

Alternatively, perhaps I can compute $ \sqrt[3]{2} $ more accurately. Let me recall that $ 1.26^3 ≈ 2.000376 $, which is very close to 2. Therefore, $ \sqrt[3]{2} $ is slightly less than 1.26. Let me compute $ 1.2599^3 $.

Let me compute $ 1.2599^3 $. Let me use linear approximation or maybe compute step by step.

Alternatively, perhaps I can use Newton-Raphson method to approximate $ \sqrt[3]{2} $. Let me try that.

Let me denote $ f(x) = x^3 - 2 $. We want to find the root of $ f(x) = 0 $. Starting with an initial guess $ x_0 = 1.26 $, since $ f(1.26) ≈ 2.000376 - 2 = 0.000376 $.

The Newton-Raphson formula is $ x_{n+1} = x_n - f(x_n)/f'(x_n) $. Here, $ f'(x) = 3x^2 $.

So, $ x_1 = 1.26 - (0.000376)/(3*(1.26)^2) $. Let me compute denominator: 3*(1.26)^2.

First, 1.26^2 = 1.5876. Multiply by 3: 4.7628.

So, $ x_1 = 1.26 - 0.000376 / 4.7628 ≈ 1.26 - 0.0000789 ≈ 1.2599211 $.

Therefore, $ \sqrt[3]{2} ≈ 1.2599211 $. Therefore, $ x_1 = \sqrt[3]{2} - 1 ≈ 0.2599211 $.

Therefore, reciprocal of $ x_1 $ is $ 1 / 0.2599211 ≈ 3.847... $. Let me compute that more accurately.

Let me compute 1 / 0.2599211. Let me note that 0.2599211 is approximately 0.259921. Let me compute 1 / 0.259921.

Let me note that 1 / 0.25 = 4, and 1 / 0.26 ≈ 3.8461538. Since 0.259921 is between 0.25 and 0.26, closer to 0.26. Let me compute 1 / 0.259921.

Let me use the linear approximation. Let me take x = 0.26, and let me consider f(x) = 1/x. Let me take x = 0.26 - Δ, where Δ = 0.26 - 0.259921 = 0.000079.

So, f(x) = 1/(0.26 - Δ) ≈ 1/0.26 + (Δ)/(0.26)^2.

Wait, using the expansion 1/(a - Δ) ≈ 1/a + Δ / a² for small Δ.

So, here, a = 0.26, Δ = 0.000079.

Therefore, 1/(0.26 - 0.000079) ≈ 1/0.26 + 0.000079 / (0.26)^2.

Compute 1/0.26 ≈ 3.8461538.

Compute 0.000079 / (0.0676) ≈ 0.000079 / 0.0676 ≈ 0.001168.

Therefore, total approximation is 3.8461538 + 0.001168 ≈ 3.84732.

Therefore, reciprocal of x1 is approximately 3.84732. Therefore, the integer part is 3, so b = 3.

But to be thorough, let me check if the actual value of 1 / x1 is more than 3.847... or less?

Alternatively, since x1 ≈ 0.259921, let me compute 3.847 * x1 ≈ 3.847 * 0.259921. Let me compute that:

First, 3 * 0.259921 = 0.779763

0.8 * 0.259921 = 0.2079368

0.04 * 0.259921 = 0.01039684

0.007 * 0.259921 ≈ 0.001819447

Adding them up: 0.779763 + 0.2079368 = 0.9876998 + 0.01039684 ≈ 0.9980966 + 0.001819447 ≈ 0.999916.

So, 3.847 * x1 ≈ 0.999916, which is very close to 1. Therefore, 1 / x1 ≈ 3.847 + (1 - 0.999916)/x1. Wait, maybe this is getting too convoluted.

Alternatively, since 3.847 * x1 ≈ 0.9999, which is just slightly less than 1. Therefore, 1 / x1 ≈ 3.847 + (1 - 0.9999)/x1 ≈ 3.847 + 0.0001 / 0.259921 ≈ 3.847 + 0.000385 ≈ 3.847385.

Therefore, the reciprocal is approximately 3.847385. Therefore, the integer part is 3, so b = 3.

But to be absolutely sure, maybe I need to compute more accurately. Let me try to compute 1 / 0.2599211.

Let me use the Newton-Raphson method for reciprocal. Let me suppose I want to compute 1 / x where x = 0.2599211. Let me take an initial guess y₀ = 3.847.

Compute y₀ * x = 3.847 * 0.2599211 ≈ 0.9999 (as above). Let me compute more accurately.

Let me compute 3.847 * 0.2599211:

First, 3 * 0.2599211 = 0.7797633

0.8 * 0.2599211 = 0.20793688

0.04 * 0.2599211 = 0.010396844

0.007 * 0.2599211 = 0.0018194477

Adding them up:

0.7797633 + 0.20793688 = 0.98770018

+ 0.010396844 = 0.998097024

+ 0.0018194477 ≈ 0.99991647

So, y₀ * x ≈ 0.99991647. Therefore, the error is 1 - 0.99991647 = 0.00008353.

Now, using Newton-Raphson for reciprocal: if we have an approximation y₀ for 1/x, then a better approximation is y₁ = y₀ * (2 - x * y₀).

So, compute y₁ = y₀ * (2 - x * y₀).

We have x * y₀ ≈ 0.99991647, so 2 - x * y₀ ≈ 2 - 0.99991647 = 1.00008353.

Therefore, y₁ ≈ 3.847 * 1.00008353 ≈ 3.847 + 3.847 * 0.00008353 ≈ 3.847 + 0.000321 ≈ 3.847321.

Therefore, y₁ ≈ 3.847321.

Now compute y₁ * x ≈ 3.847321 * 0.2599211. Let me compute this:

Again, breaking down:

3 * 0.2599211 = 0.7797633

0.8 * 0.2599211 = 0.20793688

0.04 * 0.2599211 = 0.010396844

0.007 * 0.2599211 = 0.0018194477

0.000321 * 0.2599211 ≈ 0.0000835

Adding up the previous parts: 0.7797633 + 0.20793688 = 0.98770018 + 0.010396844 = 0.998097024 + 0.0018194477 ≈ 0.99991647 + 0.0000835 ≈ 1.0000.

Therefore, y₁ * x ≈ 1.0000. Therefore, the reciprocal is approximately 3.847321. Therefore, the integer part is 3, so b = 3.

Therefore, the value of b is 3.

But wait, let me check if there's a possibility that the continued fraction might have a different value. Maybe I need to check with more precise calculation or refer to known continued fraction expansions.

Alternatively, perhaps I can use the continued fraction algorithm step by step. Let me try that.

Let me denote $ x_0 = \sqrt[3]{2} \approx 1.259921049894873 $.

First step: $ a_0 = \lfloor x_0 \rfloor = 1 $. Then, $ x_1 = 1 / (x_0 - a_0) = 1 / (x_0 - 1) \approx 1 / 0.259921049894873 ≈ 3.847322101 $.

So, $ a_1 = \lfloor x_1 \rfloor = 3 $. Then, $ x_2 = 1 / (x_1 - a_1) = 1 / (3.847322101 - 3) = 1 / 0.847322101 ≈ 1.1806... $.

So, $ a_2 = \lfloor x_2 \rfloor = 1 $. Then, $ x_3 = 1 / (x_2 - a_2) = 1 / (1.1806... - 1) = 1 / 0.1806... ≈ 5.537... $.

So, $ a_3 = 5 $. Then, $ x_4 = 1 / (x_3 - a_3) = 1 / (5.537... - 5) = 1 / 0.537... ≈ 1.862... $.

So, $ a_4 = 1 $. Then, $ x_5 = 1 / (x_4 - a_4) = 1 / (1.862... - 1) = 1 / 0.862... ≈ 1.159... $.

So, $ a_5 = 1 $. Then, $ x_6 = 1 / (x_5 - a_5) = 1 / (1.159... - 1) = 1 / 0.159... ≈ 6.289... $.

So, $ a_6 = 6 $. Then, $ x_7 = 1 / (x_6 - a_6) = 1 / (6.289... - 6) = 1 / 0.289... ≈ 3.46... $.

So, $ a_7 = 3 $. Then, $ x_8 = 1 / (x_7 - a_7) = 1 / (3.46... - 3) = 1 / 0.46... ≈ 2.173... $.

So, $ a_8 = 2 $. Then, $ x_9 = 1 / (x_8 - a_8) = 1 / (2.173... - 2) = 1 / 0.173... ≈ 5.77... $.

So, $ a_9 = 5 $. Then, $ x_{10} = 1 / (x_9 - a_9) = 1 / (5.77... - 5) = 1 / 0.77... ≈ 1.298... $.

So, $ a_{10} = 1 $. Then, $ x_{11} = 1 / (x_{10} - a_{10}) = 1 / (1.298... - 1) = 1 / 0.298... ≈ 3.355... $.

So, $ a_{11} = 3 $. Then, $ x_{12} = 1 / (x_{11} - a_{11}) = 1 / (3.355... - 3) = 1 / 0.355... ≈ 2.816... $.

So, $ a_{12} = 2 $. Then, $ x_{13} = 1 / (x_{12} - a_{12}) = 1 / (2.816... - 2) = 1 / 0.816... ≈ 1.225... $.

So, $ a_{13} = 1 $. Then, $ x_{14} = 1 / (x_{13} - a_{13}) = 1 / (1.225... - 1) = 1 / 0.225... ≈ 4.444... $.

So, $ a_{14} = 4 $. Then, $ x_{15} = 1 / (x_{14} - a_{14}) = 1 / (4.444... - 4) = 1 / 0.444... ≈ 2.25... $.

So, $ a_{15} = 2 $. Then, $ x_{16} = 1 / (x_{15} - a_{15}) = 1 / (2.25... - 2) = 1 / 0.25... = 4 $.

So, $ a_{16} = 4 $. Then, $ x_{17} = 1 / (x_{16} - a_{16}) = 1 / (4 - 4) = undefined. Wait, but that can't be. Wait, x_{16} is 4, so x_{16} - a_{16} = 0. So, division by zero? That suggests that the continued fraction would terminate here? But since $ \sqrt[3]{2} $ is irrational, the continued fraction should be infinite. Therefore, my approximations must be getting too rough here.

But in reality, the continued fraction of an irrational number is infinite, so my approximations are just getting closer and closer, but not exact. Therefore, perhaps my manual calculations are introducing errors.

But regardless, the first few terms of the continued fraction are:

$ a_0 = 1 $, $ a_1 = 3 $, $ a_2 = 1 $, $ a_3 = 5 $, $ a_4 = 1 $, $ a_5 = 1 $, $ a_6 = 6 $, $ a_7 = 3 $, $ a_8 = 2 $, $ a_9 = 5 $, $ a_{10} = 1 $, $ a_{11} = 3 $, $ a_{12} = 2 $, $ a_{13} = 1 $, $ a_{14} = 4 $, $ a_{15} = 2 $, $ a_{16} = 4 $, etc.

But since the question only asks for $ b $, which is the second term in the continued fraction. Wait, the original expression is:

$$
\sqrt[3]{2} = a + \cfrac{1}{b + \cfrac{1}{c + \cfrac{1}{d + \dotsb}}}
$$

So, in the standard continued fraction notation, the continued fraction is $ [a_0; a_1, a_2, a_3, ...] $. Therefore, in this case, $ a = a_0 = 1 $, and $ b = a_1 = 3 $. Therefore, the answer is 3.

But to make sure, let me check if there is any source or known continued fraction for cube roots.

Alternatively, maybe I can check with another method. Let me consider that the continued fraction expansion of $ \sqrt[3]{2} $ is known. However, I don't remember it off the top of my head.

Alternatively, perhaps I can use the fact that continued fractions are unique for irrational numbers, so if my calculation shows that the first term after the integer part is 3, then that should be correct.

Alternatively, let me check with another approach. Let me suppose that $ \sqrt[3]{2} = 1 + 1/(3 + 1/(c + ...)) $. Let me compute the value of $ 1 + 1/(3 + 1/(c + ...)) $ and see if it's close to $ \sqrt[3]{2} $.

Let me compute $ 1 + 1/3 = 1.333... $. But $ \sqrt[3]{2} ≈ 1.2599 $, which is less than 1.333. Therefore, the next term in the continued fraction must be such that $ 1 + 1/(3 + 1/c + ...) $ is less than 1.333. Therefore, the next term is 1/(something larger than 3).

Wait, but in the continued fraction, after the first term, we have $ 1/(b + ...) $. So, if $ b = 3 $, then the next term is $ 1/(3 + ...) $, so the total value is $ 1 + 1/(3 + ...) $.

But if I take $ 1 + 1/(3 + 1/c) $, then let me compute that. Let me assume that the continued fraction is $ 1 + 1/(3 + 1/c) $. Let me compute this value.

Let me suppose that the continued fraction is finite for the sake of calculation. Let me take c as some integer. Let me compute $ 1 + 1/(3 + 1/c) $.

Let me compute for c = 1: $ 1 + 1/(3 + 1/1) = 1 + 1/4 = 1.25 $.

Compare with $ \sqrt[3]{2} ≈ 1.2599 $. So, 1.25 is less than that.

If c = 2: $ 1 + 1/(3 + 1/2) = 1 + 1/(3.5) ≈ 1 + 0.2857 ≈ 1.2857 $. Which is higher than 1.2599.

Therefore, the actual value of c must be between 1 and 2? But c is a positive integer, so c must be 1 or 2. But since the value of the continued fraction with c=1 gives 1.25, which is less than $ \sqrt[3]{2} $, and with c=2 gives 1.2857, which is higher. Therefore, the actual continued fraction must have c=1, and then the next term would be something else.

Wait, but in reality, the continued fraction is infinite, so we can't just truncate it. However, this suggests that the continued fraction after 1 + 1/3 is something like 1 + 1/(3 + 1/(1 + ...)).

But according to my previous calculation, the next term after 3 is 1, so c=1.

But let me check with the value. Let me compute the continued fraction up to the term c=1.

So, if we take the continued fraction as $ 1 + 1/(3 + 1/1) = 1 + 1/4 = 1.25 $. But $ \sqrt[3]{2} ≈ 1.2599 $, so the difference is about 0.0099.

If we take the next term, which is 5, as per my previous calculation. Let me compute $ 1 + 1/(3 + 1/(1 + 1/5)) $.

Compute denominator: 3 + 1/(1 + 1/5) = 3 + 1/(6/5) = 3 + 5/6 ≈ 3.8333. Then reciprocal is 1 / 3.8333 ≈ 0.2608. Then add 1: 1.2608. Which is very close to $ \sqrt[3]{2} ≈ 1.2599 $. The difference is about 0.0009.

So, that's a much better approximation. Therefore, the continued fraction up to that point is [1; 3, 1, 5], which gives approximately 1.2608, very close to the actual value.

Therefore, this suggests that the continued fraction starts with 1; 3, 1, 5,..., so the value of b is 3.

Therefore, I think my initial calculation is correct.

But just to make sure, let me check with another approach. Let me consider the continued fraction expansion of $ \sqrt[3]{2} $.

Alternatively, maybe I can use the fact that continued fractions can be calculated using the Euclidean algorithm, but for cube roots, it's more complicated.

Alternatively, perhaps I can use the fact that $ \sqrt[3]{2} $ satisfies the equation $ x^3 = 2 $. Maybe I can use this equation to find relations in the continued fraction.

But I don't know how to directly relate that to the continued fraction terms.

Alternatively, maybe I can use the convergents of the continued fraction.

The convergents are the fractions obtained by truncating the continued fraction. For example, the first convergent is 1, the second is 1 + 1/3 = 4/3 ≈ 1.333..., the third convergent is 1 + 1/(3 + 1/1) = 1 + 1/(4) = 5/4 = 1.25, the fourth convergent is 1 + 1/(3 + 1/(1 + 1/5)) = 1 + 1/(3 + 1/(6/5)) = 1 + 1/(3 + 5/6) = 1 + 1/(23/6) = 1 + 6/23 = 29/23 ≈ 1.26087, which is very close to $ \sqrt[3]{2} ≈ 1.25992 $.

So, the convergents are approaching the actual value. Therefore, the continued fraction is [1; 3, 1, 5, 1, 1, 6, 3, 2, 5, 1, 3, 2, 1, 4, 2, 4,...], as per my previous calculation.

Therefore, the second term is 3, so b = 3.

Therefore, after all these calculations and checks, I can be confident that the value of $ b $ is 3.

**Final Answer**
\boxed{3}

To determine the value of $ b $ in the continued fraction expansion of $ \sqrt[3]{2} $, we begin by understanding the structure of the given expression:

$$
\sqrt[3]{2} = a + \cfrac{1}{b + \cfrac{1}{c + \cfrac{1}{d + \dotsb}}}
$$

This is a **continued fraction** representation of the cube root of 2, where $ a, b, c, d, \dots $ are positive integers. Our goal is to find the value of $ b $, the second term in the continued fraction.

---

### Step 1: Approximate $ \sqrt[3]{2} $

We know that:

$$
1^3 = 1, \quad 2^3 = 8 \Rightarrow \sqrt[3]{2} \in (1, 2)
$$

Using a more precise approximation:

$$
\sqrt[3]{2} \approx 1.259921049894873
$$

So, the **integer part** $ a $ is:

$$
a = \lfloor \sqrt[3]{2} \rfloor = 1
$$

---

### Step 2: Compute the Continued Fraction

We now compute the continued fraction by repeatedly taking the reciprocal of the fractional part and extracting the integer part.

Let:

$$
x_0 = \sqrt[3]{2} \approx 1.259921049894873
$$

Then:

$$
x_1 = \frac{1}{x_0 - a} = \frac{1}{\sqrt[3]{2} - 1} \approx \frac{1}{0.259921049894873} \approx 3.847322101
$$

So, the next term $ b $ is:

$$
b = \lfloor x_1 \rfloor = 3
$$

---

### Step 3: Verify the Continued Fraction

To confirm, we can compute the next few terms of the continued fraction:

- $ x_2 = \frac{1}{x_1 - b} = \frac{1}{3.847322101 - 3} = \frac{1}{0.847322101} \approx 1.1806 $
- $ x_3 = \frac{1}{x_2 - 1} = \frac{1}{0.1806} \approx 5.537 $
- $ x_4 = \frac{1}{x_3 - 5} = \frac{1}{0.537} \approx 1.862 $
- $ x_5 = \frac{1}{x_4 - 1} = \frac{1}{0.862} \approx 1.159 $
- $ x_6 = \frac{1}{x_5 - 1} = \frac{1}{0.159} \approx 6.289 $

This confirms that the continued fraction expansion of $ \sqrt[3]{2} $ is:

$$
\sqrt[3]{2} = 1 + \cfrac{1}{3 + \cfrac{1}{1 + \cfrac{1}{5 + \cfrac{1}{1 + \dotsb}}}}
$$

Thus, the second term in the continued fraction is:

$$
\boxed{3}
$$

[Assistant]
Explanation:
Okay, so I need to find the value of $ b $ in the continued fraction expansion of $ \sqrt[3]{2} $. The given expression is:

$$
\sqrt[3]{2} = a + \cfrac{1}{b + \cfrac{1}{c + \cfrac{1}{d + \dotsb}}}
$$

where $ a, b, c, d $ are positive integers. My task is to compute $ b $.

First, I remember that continued fractions for irrational numbers are infinite and non-repeating. Since $ \sqrt[3]{2} $ is irrational, its continued fraction will be infinite. However, I need to find the first few terms of this continued fraction, specifically the value of $ b $, which is the second term in the expansion.

Let me recall how continued fractions work. For any real number $ x $, the continued fraction expansion is found by taking the integer part, then taking the reciprocal of the fractional part, and repeating the process. So, if I can compute the continued fraction step by step, I can find $ a $, then $ b $, and so on.

Let me start by finding $ a $. Since $ \sqrt[3]{2} $ is approximately... let me calculate that.

I know that $ 1^3 = 1 $, $ 2^3 = 8 $, so $ \sqrt[3]{2} $ is between 1 and 2. Let me compute it more accurately.

Let me recall that $ \sqrt[3]{1} = 1 $, $ \sqrt[3]{8} = 2 $. Let me approximate $ \sqrt[3]{2} $.

I know that $ 1.2^3 = 1.728 $, which is less than 2. $ 1.26^3 = (1.2 + 0.06)^3 $. Let me compute that:

First, $ 1.2^3 = 1.728 $, then $ 1.26^3 $:

Let me compute $ 1.26 \times 1.26 = 1.5876 $, then multiply by 1.26:

$ 1.5876 \times 1.26 $. Let me compute that:

First, 1.5876 * 1 = 1.5876

1.5876 * 0.2 = 0.31752

1.5876 * 0.06 = 0.095256

Adding them up: 1.5876 + 0.31752 = 1.90512 + 0.095256 ≈ 2.000376.

Wow, so $ 1.26^3 ≈ 2.000376 $, which is very close to 2. Therefore, $ \sqrt[3]{2} $ is approximately 1.26. Therefore, the integer part $ a $ is 1.

So, $ a = 1 $. Therefore, we can write:

$$
\sqrt[3]{2} = 1 + \cfrac{1}{b + \cfrac{1}{c + \cfrac{1}{d + \dotsb}}}
$$

So, to find $ b $, I need to compute the continued fraction of $ \sqrt[3]{2} - 1 $, which is approximately 0.26. Let me denote $ x_1 = \sqrt[3]{2} - 1 \approx 0.26 $.

The continued fraction process is as follows: take the reciprocal of $ x_1 $, take the integer part as the next term, and repeat.

So, let me compute $ x_1 = \sqrt[3]{2} - 1 \approx 0.259921049894873... $.

So, $ x_1 \approx 0.259921 $.

Then, the reciprocal of $ x_1 $ is $ 1 / x_1 \approx 1 / 0.259921 ≈ 3.847... $. Therefore, the integer part is 3, so $ b = 3 $. Wait, but let me check this more carefully.

But before I jump to conclusions, maybe I need to do this more accurately. Because if I approximate $ \sqrt[3]{2} $ as 1.26, then $ x_1 = 0.26 $, reciprocal is 3.846... So, the integer part is 3, so $ b = 3 $. But maybe the exact value of $ x_1 $ is slightly different, so maybe the reciprocal is slightly different? Let me check with more accurate value of $ \sqrt[3]{2} $.

Alternatively, perhaps I can compute $ \sqrt[3]{2} $ more accurately. Let me recall that $ 1.26^3 ≈ 2.000376 $, which is very close to 2. Therefore, $ \sqrt[3]{2} $ is slightly less than 1.26. Let me compute $ 1.2599^3 $.

Let me compute $ 1.2599^3 $. Let me use linear approximation or maybe compute step by step.

Alternatively, perhaps I can use Newton-Raphson method to approximate $ \sqrt[3]{2} $. Let me try that.

Let me denote $ f(x) = x^3 - 2 $. We want to find the root of $ f(x) = 0 $. Starting with an initial guess $ x_0 = 1.26 $, since $ f(1.26) ≈ 2.000376 - 2 = 0.000376 $.

The Newton-Raphson formula is $ x_{n+1} = x_n - f(x_n)/f'(x_n) $. Here, $ f'(x) = 3x^2 $.

So, $ x_1 = 1.26 - (0.000376)/(3*(1.26)^2) $. Let me compute denominator: 3*(1.26)^2.

First, 1.26^2 = 1.5876. Multiply by 3: 4.7628.

So, $ x_1 = 1.26 - 0.000376 / 4.7628 ≈ 1.26 - 0.0000789 ≈ 1.2599211 $.

Therefore, $ \sqrt[3]{2} ≈ 1.2599211 $. Therefore, $ x_1 = \sqrt[3]{2} - 1 ≈ 0.2599211 $.

Therefore, reciprocal of $ x_1 $ is $ 1 / 0.2599211 ≈ 3.847... $. Let me compute that more accurately.

Let me compute 1 / 0.2599211. Let me note that 0.2599211 is approximately 0.259921. Let me compute 1 / 0.259921.

Let me note that 1 / 0.25 = 4, and 1 / 0.26 ≈ 3.8461538. Since 0.259921 is between 0.25 and 0.26, closer to 0.26. Let me compute 1 / 0.259921.

Let me use the linear approximation. Let me take x = 0.26, and let me consider f(x) = 1/x. Let me take x = 0.26 - Δ, where Δ = 0.26 - 0.259921 = 0.000079.

So, f(x) = 1/(0.26 - Δ) ≈ 1/0.26 + (Δ)/(0.26)^2.

Wait, using the expansion 1/(a - Δ) ≈ 1/a + Δ / a² for small Δ.

So, here, a = 0.26, Δ = 0.000079.

Therefore, 1/(0.26 - 0.000079) ≈ 1/0.26 + 0.000079 / (0.26)^2.

Compute 1/0.26 ≈ 3.8461538.

Compute 0.000079 / (0.0676) ≈ 0.000079 / 0.0676 ≈ 0.001168.

Therefore, total approximation is 3.8461538 + 0.001168 ≈ 3.84732.

Therefore, reciprocal of x1 is approximately 3.84732. Therefore, the integer part is 3, so b = 3.

But to be thorough, let me check if the actual value of 1 / x1 is more than 3.847... or less?

Alternatively, since x1 ≈ 0.259921, let me compute 3.847 * x1 ≈ 3.847 * 0.259921. Let me compute that:

First, 3 * 0.259921 = 0.779763

0.8 * 0.259921 = 0.2079368

0.04 * 0.259921 = 0.01039684

0.007 * 0.259921 ≈ 0.001819447

Adding them up: 0.779763 + 0.2079368 = 0.9876998 + 0.01039684 ≈ 0.9980966 + 0.001819447 ≈ 0.999916.

So, 3.847 * x1 ≈ 0.999916, which is very close to 1. Therefore, 1 / x1 ≈ 3.847 + (1 - 0.999916)/x1. Wait, maybe this is getting too convoluted.

Alternatively, since 3.847 * x1 ≈ 0.9999, which is just slightly less than 1. Therefore, 1 / x1 ≈ 3.847 + (1 - 0.9999)/x1 ≈ 3.847 + 0.0001 / 0.259921 ≈ 3.847 + 0.000385 ≈ 3.847385.

Therefore, the reciprocal is approximately 3.847385. Therefore, the integer part is 3, so b = 3.

But to be absolutely sure, maybe I need to compute more accurately. Let me try to compute 1 / 0.2599211.

Let me use the Newton-Raphson method for reciprocal. Let me suppose I want to compute 1 / x where x = 0.2599211. Let me take an initial guess y₀ = 3.847.

Compute y₀ * x = 3.847 * 0.2599211 ≈ 0.9999 (as above). Let me compute more accurately.

Let me compute 3.847 * 0.2599211:

First, 3 * 0.2599211 = 0.7797633

0.8 * 0.2599211 = 0.20793688

0.04 * 0.2599211 = 0.010396844

0.007 * 0.2599211 = 0.0018194477

Adding them up:

0.7797633 + 0.20793688 = 0.98770018

+ 0.010396844 = 0.998097024

+ 0.0018194477 ≈ 0.99991647

So, y₀ * x ≈ 0.99991647. Therefore, the error is 1 - 0.99991647 = 0.00008353.

Now, using Newton-Raphson for reciprocal: if we have an approximation y₀ for 1/x, then a better approximation is y₁ = y₀ * (2 - x * y₀).

So, compute y₁ = y₀ * (2 - x * y₀).

We have x * y₀ ≈ 0.99991647, so 2 - x * y₀ ≈ 2 - 0.99991647 = 1.00008353.

Therefore, y₁ ≈ 3.847 * 1.00008353 ≈ 3.847 + 3.847 * 0.00008353 ≈ 3.847 + 0.000321 ≈ 3.847321.

Therefore, y₁ ≈ 3.847321.

Now compute y₁ * x ≈ 3.847321 * 0.2599211. Let me compute this:

Again, breaking down:

3 * 0.2599211 = 0.7797633

0.8 * 0.2599211 = 0.20793688

0.04 * 0.2599211 = 0.010396844

0.007 * 0.2599211 = 0.0018194477

0.000321 * 0.2599211 ≈ 0.0000835

Adding up the previous parts: 0.7797633 + 0.20793688 = 0.98770018 + 0.010396844 = 0.998097024 + 0.0018194477 ≈ 0.99991647 + 0.0000835 ≈ 1.0000.

Therefore, y₁ * x ≈ 1.0000. Therefore, the reciprocal is approximately 3.847321. Therefore, the integer part is 3, so b = 3.

Therefore, the value of b is 3.

But wait, let me check if there's a possibility that the continued fraction might have a different value. Maybe I need to check with more precise calculation or refer to known continued fraction expansions.

Alternatively, perhaps I can use the continued fraction algorithm step by step. Let me try that.

Let me denote $ x_0 = \sqrt[3]{2} \approx 1.259921049894873 $.

First step: $ a_0 = \lfloor x_0 \rfloor = 1 $. Then, $ x_1 = 1 / (x_0 - a_0) = 1 / (x_0 - 1) \approx 1 / 0.259921049894873 ≈ 3.847322101 $.

So, $ a_1 = \lfloor x_1 \rfloor = 3 $. Then, $ x_2 = 1 / (x_1 - a_1) = 1 / (3.847322101 - 3) = 1 / 0.847322101 ≈ 1.1806... $.

So, $ a_2 = \lfloor x_2 \rfloor = 1 $. Then, $ x_3 = 1 / (x_2 - a_2) = 1 / (1.1806... - 1) = 1 / 0.1806... ≈ 5.537... $.

So, $ a_3 = 5 $. Then, $ x_4 = 1 / (x_3 - a_3) = 1 / (5.537... - 5) = 1 / 0.537... ≈ 1.862... $.

So, $ a_4 = 1 $. Then, $ x_5 = 1 / (x_4 - a_4) = 1 / (1.862... - 1) = 1 / 0.862... ≈ 1.159... $.

So, $ a_5 = 1 $. Then, $ x_6 = 1 / (x_5 - a_5) = 1 / (1.159... - 1) = 1 / 0.159... ≈ 6.289... $.

So, $ a_6 = 6 $. Then, $ x_7 = 1 / (x_6 - a_6) = 1 / (6.289... - 6) = 1 / 0.289... ≈ 3.46... $.

So, $ a_7 = 3 $. Then, $ x_8 = 1 / (x_7 - a_7) = 1 / (3.46... - 3) = 1 / 0.46... ≈ 2.173... $.

So, $ a_8 = 2 $. Then, $ x_9 = 1 / (x_8 - a_8) = 1 / (2.173... - 2) = 1 / 0.173... ≈ 5.77... $.

So, $ a_9 = 5 $. Then, $ x_{10} = 1 / (x_9 - a_9) = 1 / (5.77... - 5) = 1 / 0.77... ≈ 1.298... $.

So, $ a_{10} = 1 $. Then, $ x_{11} = 1 / (x_{10} - a_{10}) = 1 / (1.298... - 1) = 1 / 0.298... ≈ 3.355... $.

So, $ a_{11} = 3 $. Then, $ x_{12} = 1 / (x_{11} - a_{11}) = 1 / (3.355... - 3) = 1 / 0.355... ≈ 2.816... $.

So, $ a_{12} = 2 $. Then, $ x_{13} = 1 / (x_{12} - a_{12}) = 1 / (2.816... - 2) = 1 / 0.816... ≈ 1.225... $.

So, $ a_{13} = 1 $. Then, $ x_{14} = 1 / (x_{13} - a_{13}) = 1 / (1.225... - 1) = 1 / 0.225... ≈ 4.444... $.

So, $ a_{14} = 4 $. Then, $ x_{15} = 1 / (x_{14} - a_{14}) = 1 / (4.444... - 4) = 1 / 0.444... ≈ 2.25... $.

So, $ a_{15} = 2 $. Then, $ x_{16} = 1 / (x_{15} - a_{15}) = 1 / (2.25... - 2) = 1 / 0.25... = 4 $.

So, $ a_{16} = 4 $. Then, $ x_{17} = 1 / (x_{16} - a_{16}) = 1 / (4 - 4) = undefined. Wait, but that can't be. Wait, x_{16} is 4, so x_{16} - a_{16} = 0. So, division by zero? That suggests that the continued fraction would terminate here? But since $ \sqrt[3]{2} $ is irrational, the continued fraction should be infinite. Therefore, my approximations must be getting too rough here.

But in reality, the continued fraction of an irrational number is infinite, so my approximations are just getting closer and closer, but not exact. Therefore, perhaps my manual calculations are introducing errors.

But regardless, the first few terms of the continued fraction are:

$ a_0 = 1 $, $ a_1 = 3 $, $ a_2 = 1 $, $ a_3 = 5 $, $ a_4 = 1 $, $ a_5 = 1 $, $ a_6 = 6 $, $ a_7 = 3 $, $ a_8 = 2 $, $ a_9 = 5 $, $ a_{10} = 1 $, $ a_{11} = 3 $, $ a_{12} = 2 $, $ a_{13} = 1 $, $ a_{14} = 4 $, $ a_{15} = 2 $, $ a_{16} = 4 $, etc.

But since the question only asks for $ b $, which is the second term in the continued fraction. Wait, the original expression is:

$$
\sqrt[3]{2} = a + \cfrac{1}{b + \cfrac{1}{c + \cfrac{1}{d + \dotsb}}}
$$

So, in the standard continued fraction notation, the continued fraction is $ [a_0; a_1, a_2, a_3, ...] $. Therefore, in this case, $ a = a_0 = 1 $, and $ b = a_1 = 3 $. Therefore, the answer is 3.

But to make sure, let me check if there is any source or known continued fraction for cube roots.

Alternatively, maybe I can check with another method. Let me consider that the continued fraction expansion of $ \sqrt[3]{2} $ is known. However, I don't remember it off the top of my head.

Alternatively, perhaps I can use the fact that continued fractions are unique for irrational numbers, so if my calculation shows that the first term after the integer part is 3, then that should be correct.

Alternatively, let me check with another approach. Let me suppose that $ \sqrt[3]{2} = 1 + 1/(3 + 1/(c + ...)) $. Let me compute the value of $ 1 + 1/(3 + 1/(c + ...)) $ and see if it's close to $ \sqrt[3]{2} $.

Let me compute $ 1 + 1/3 = 1.333... $. But $ \sqrt[3]{2} ≈ 1.2599 $, which is less than 1.333. Therefore, the next term in the continued fraction must be such that $ 1 + 1/(3 + 1/c + ...) $ is less than 1.333. Therefore, the next term is 1/(something larger than 3).

Wait, but in the continued fraction, after the first term, we have $ 1/(b + ...) $. So, if $ b = 3 $, then the next term is $ 1/(3 + ...) $, so the total value is $ 1 + 1/(3 + ...) $.

But if I take $ 1 + 1/(3 + 1/c) $, then let me compute that. Let me assume that the continued fraction is $ 1 + 1/(3 + 1/c) $. Let me compute this value.

Let me suppose that the continued fraction is finite for the sake of calculation. Let me take c as some integer. Let me compute $ 1 + 1/(3 + 1/c) $.

Let me compute for c = 1: $ 1 + 1/(3 + 1/1) = 1 + 1/4 = 1.25 $.

Compare with $ \sqrt[3]{2} ≈ 1.2599 $. So, 1.25 is less than that.

If c = 2: $ 1 + 1/(3 + 1/2) = 1 + 1/(3.5) ≈ 1 + 0.2857 ≈ 1.2857 $. Which is higher than 1.2599.

Therefore, the actual value of c must be between 1 and 2? But c is a positive integer, so c must be 1 or 2. But since the value of the continued fraction with c=1 gives 1.25, which is less than $ \sqrt[3]{2} $, and with c=2 gives 1.2857, which is higher. Therefore, the actual continued fraction must have c=1, and then the next term would be something else.

Wait, but in reality, the continued fraction is infinite, so we can't just truncate it. However, this suggests that the continued fraction after 1 + 1/3 is something like 1 + 1/(3 + 1/(1 + ...)).

But according to my previous calculation, the next term after 3 is 1, so c=1.

But let me check with the value. Let me compute the continued fraction up to the term c=1.

So, if we take the continued fraction as $ 1 + 1/(3 + 1/1) = 1 + 1/4 = 1.25 $. But $ \sqrt[3]{2} ≈ 1.2599 $, so the difference is about 0.0099.

If we take the next term, which is 5, as per my previous calculation. Let me compute $ 1 + 1/(3 + 1/(1 + 1/5)) $.

Compute denominator: 3 + 1/(1 + 1/5) = 3 + 1/(6/5) = 3 + 5/6 ≈ 3.8333. Then reciprocal is 1 / 3.8333 ≈ 0.2608. Then add 1: 1.2608. Which is very close to $ \sqrt[3]{2} ≈ 1.2599 $. The difference is about 0.0009.

So, that's a much better approximation. Therefore, the continued fraction up to that point is [1; 3, 1, 5], which gives approximately 1.2608, very close to the actual value.

Therefore, this suggests that the continued fraction starts with 1; 3, 1, 5,..., so the value of b is 3.

Therefore, I think my initial calculation is correct.

But just to make sure, let me check with another approach. Let me consider the continued fraction expansion of $ \sqrt[3]{2} $.

Alternatively, maybe I can use the fact that continued fractions can be calculated using the Euclidean algorithm, but for cube roots, it's more complicated.

Alternatively, perhaps I can use the fact that $ \sqrt[3]{2} $ satisfies the equation $ x^3 = 2 $. Maybe I can use this equation to find relations in the continued fraction.

But I don't know how to directly relate that to the continued fraction terms.

Alternatively, maybe I can use the convergents of the continued fraction.

The convergents are the fractions obtained by truncating the continued fraction. For example, the first convergent is 1, the second is 1 + 1/3 = 4/3 ≈ 1.333..., the third convergent is 1 + 1/(3 + 1/1) = 1 + 1/(4) = 5/4 = 1.25, the fourth convergent is 1 + 1/(3 + 1/(1 + 1/5)) = 1 + 1/(3 + 1/(6/5)) = 1 + 1/(3 + 5/6) = 1 + 1/(23/6) = 1 + 6/23 = 29/23 ≈ 1.26087, which is very close to $ \sqrt[3]{2} ≈ 1.25992 $.

So, the convergents are approaching the actual value. Therefore, the continued fraction is [1; 3, 1, 5, 1, 1, 6, 3, 2, 5, 1, 3, 2, 1, 4, 2, 4,...], as per my previous calculation.

Therefore, the second term is 3, so b = 3.

Therefore, after all these calculations and checks, I can be confident that the value of $ b $ is 3.

**Final Answer**
\boxed{3}

To determine the value of $ b $ in the continued fraction expansion of $ \sqrt[3]{2} $, we begin by understanding the structure of the given expression:

$$
\sqrt[3]{2} = a + \cfrac{1}{b + \cfrac{1}{c + \cfrac{1}{d + \dotsb}}}
$$

This is a **continued fraction** representation of the cube root of 2, where $ a, b, c, d, \dots $ are positive integers. Our goal is to find the value of $ b $, the second term in the continued fraction.

---

### Step 1: Approximate $ \sqrt[3]{2} $

We know that:

$$
1^3 = 1, \quad 2^3 = 8 \Rightarrow \sqrt[3]{2} \in (1, 2)
$$

Using a more precise approximation:

$$
\sqrt[3]{2} \approx 1.259921049894873
$$

So, the **integer part** $ a $ is:

$$
a = \lfloor \sqrt[3]{2} \rfloor = 1
$$

---

### Step 2: Compute the Continued Fraction

We now compute the continued fraction by repeatedly taking the reciprocal of the fractional part and extracting the integer part.

Let:

$$
x_0 = \sqrt[3]{2} \approx 1.259921049894873
$$

Then:

$$
x_1 = \frac{1}{x_0 - a} = \frac{1}{\sqrt[3]{2} - 1} \approx \frac{1}{0.259921049894873} \approx 3.847322101
$$

So, the next term $ b $ is:

$$
b = \lfloor x_1 \rfloor = 3
$$

---

### Step 3: Verify the Continued Fraction

To confirm, we can compute the next few terms of the continued fraction:

- $ x_2 = \frac{1}{x_1 - b} = \frac{1}{3.847322101 - 3} = \frac{1}{0.847322101} \approx 1.1806 $
- $ x_3 = \frac{1}{x_2 - 1} = \frac{1}{0.1806} \approx 5.537 $
- $ x_4 = \frac{1}{x_3 - 5} = \frac{1}{0.537} \approx 1.862 $
- $ x_5 = \frac{1}{x_4 - 1} = \frac{1}{0.862} \approx 1.159 $
- $ x_6 = \frac{1}{x_5 - 1} = \frac{1}{0.159} \approx 6.289 $

This confirms that the continued fraction expansion of $ \sqrt[3]{2} $ is:

$$
\sqrt[3]{2} = 1 + \cfrac{1}{3 + \cfrac{1}{1 + \cfrac{1}{5 + \cfrac{1}{1 + \dotsb}}}}
$$

Thus, the second term in the continued fraction is:

$$
\boxed{3}
$$

Solution:
We know that
\[\sqrt[3]{2} = a + \cfrac{1}{b + \cfrac{1}{c + \cfrac{1}{d + \dotsb}}} > a,\]and
\[\sqrt[3]{2} = a + \cfrac{1}{b + \cfrac{1}{c + \cfrac{1}{d + \dotsb}}} < a + 1.\]The integer $a$ that satisfies $a < \sqrt[3]{2} < a + 1$ is $a = 1.$

Then
\[\sqrt[3]{2} - 1 = \cfrac{1}{b + \cfrac{1}{c + \cfrac{1}{d + \dotsb}}},\]so
\[\frac{1}{\sqrt[3]{2} - 1} = b + \cfrac{1}{c + \cfrac{1}{d + \dotsb}}.\]As before, $b$ must satisfy
\[b < \frac{1}{\sqrt[3]{2} - 1} < b + 1.\]Rationalizing the denominator, we get
\[\frac{1}{\sqrt[3]{2} - 1} = \frac{\sqrt[3]{4} + \sqrt[3]{2} + 1}{(\sqrt[3]{2} - 1)(\sqrt[3]{4} + \sqrt[3]{2} + 1)} = \frac{\sqrt[3]{4} + \sqrt[3]{2} + 1}{2 - 1} = \sqrt[3]{4} + \sqrt[3]{2} + 1.\]We have that
\[\sqrt[3]{4} + \sqrt[3]{2} + 1 > 1 + 1 + 1 = 3.\]Also, $1.3^3 = 2.197 > 2$ and $1.6^3 = 4.096 > 4,$ so
\[\sqrt[3]{4} + \sqrt[3]{2} + 1 < 1.3 + 1.6 + 1 = 3.9 < 4,\]so $b = \boxed{3}.$