Explanation:
Okay, so I need to find the minimum value of $ a^4 + b^8 $ given that $ ab^2 = 5 $ where $ a $ and $ b $ are positive real numbers. Hmm, let me think about how to approach this.

First, since there's a constraint $ ab^2 = 5 $, maybe I can use substitution to express one variable in terms of the other and then minimize the function. Let me try that.

Let me solve the constraint equation for $ a $. If $ ab^2 = 5 $, then $ a = \frac{5}{b^2} $. Now, substitute this into the expression we need to minimize, which is $ a^4 + b^8 $.

So substituting $ a $, we get:

$ \left( \frac{5}{b^2} \right)^4 + b^8 $

Let me compute that:

First, $ \left( \frac{5}{b^2} \right)^4 = \frac{5^4}{b^8} = \frac{625}{b^8} $. Therefore, the expression becomes:

$ \frac{625}{b^8} + b^8 $

So now, the problem reduces to finding the minimum of the function $ f(b) = \frac{625}{b^8} + b^8 $ where $ b > 0 $.

Hmm, okay. To find the minimum, I can take the derivative of $ f(b) $ with respect to $ b $, set it equal to zero, and solve for $ b $. Then check if it's a minimum using the second derivative or some other method. Let me try that.

First, compute the derivative $ f'(b) $:

$ f(b) = 625 b^{-8} + b^8 $

So, derivative term by term:

The derivative of $ 625 b^{-8} $ is $ 625 \times (-8) b^{-9} = -5000 b^{-9} $

The derivative of $ b^8 $ is $ 8 b^7 $

Therefore, $ f'(b) = -5000 b^{-9} + 8 b^7 $

Set this equal to zero for critical points:

$ -5000 b^{-9} + 8 b^7 = 0 $

Let me rewrite this equation:

$ 8 b^7 = 5000 b^{-9} $

Multiply both sides by $ b^9 $ to eliminate the negative exponent:

$ 8 b^{16} = 5000 $

Then, divide both sides by 8:

$ b^{16} = \frac{5000}{8} $

Calculate $ 5000 / 8 $:

Well, 5000 divided by 8 is 625. So:

$ b^{16} = 625 $

Now, solve for $ b $. Since $ b > 0 $, we can take the 16th root:

$ b = 625^{1/16} $

Hmm, let me simplify 625. 625 is 5^4, right? Because 5^4 = 625. So:

$ b = (5^4)^{1/16} = 5^{4/16} = 5^{1/4} $

So $ b = 5^{1/4} $. Let me check that again.

Yes, because $ 5^{1/4} $ is the fourth root of 5, which is the same as $ 5^{0.25} $.

Alternatively, since $ b^{16} = 5^4 $, taking both sides to the power of 1/16 gives $ b = 5^{4/16} = 5^{1/4} $. Correct.

So, now that I have $ b $, I can find $ a $ using the original constraint $ ab^2 = 5 $. Let me compute $ a $:

$ a = \frac{5}{b^2} $

Since $ b = 5^{1/4} $, then $ b^2 = (5^{1/4})^2 = 5^{1/2} = \sqrt{5} $. Therefore:

$ a = \frac{5}{\sqrt{5}} = \sqrt{5} $. Because $ 5 / \sqrt{5} = \sqrt{5} \times \sqrt{5} / \sqrt{5} = \sqrt{5} $.

So, $ a = \sqrt{5} $, $ b = 5^{1/4} $.

Now, I need to compute the value of $ a^4 + b^8 $ at these values to find the minimum. Let me compute each term:

First, $ a^4 $:

Since $ a = \sqrt{5} = 5^{1/2} $, then $ a^4 = (5^{1/2})^4 = 5^{2} = 25 $.

Next, $ b^8 $:

Since $ b = 5^{1/4} $, then $ b^8 = (5^{1/4})^8 = 5^{2} = 25 $.

Therefore, $ a^4 + b^8 = 25 + 25 = 50 $.

So, the minimum value is 50? Wait, but let me check if this is indeed a minimum. Maybe I should verify using the second derivative or check the behavior of the function.

Alternatively, since we found a critical point and the function tends to infinity as $ b $ approaches 0 or infinity, this critical point is likely to be the minimum. Let me confirm with the second derivative.

Compute the second derivative $ f''(b) $:

We had $ f'(b) = -5000 b^{-9} + 8 b^7 $

Differentiate again:

The derivative of $ -5000 b^{-9} $ is $ -5000 \times (-9) b^{-10} = 45000 b^{-10} $

The derivative of $ 8 b^7 $ is $ 56 b^6 $

Therefore, $ f''(b) = 45000 b^{-10} + 56 b^6 $

Since $ b > 0 $, both terms are positive. Therefore, $ f''(b) > 0 $, which means the function is convex at this critical point, so it is indeed a local minimum. Since the function tends to infinity as $ b $ approaches 0 or infinity, this local minimum is the global minimum.

Therefore, the minimum value of $ a^4 + b^8 $ is 50.

But let me think again if there's another way to approach this problem, maybe using inequalities like AM-GM? Maybe that could be another method to confirm the answer.

Let me try using the AM-GM inequality. The expression we need to minimize is $ a^4 + b^8 $. However, the constraint is $ ab^2 = 5 $.

But AM-GM usually applies to sums of terms with exponents that can be balanced. Let me see.

Alternatively, maybe I can write $ a^4 + b^8 $ as a sum of terms that can be related to the constraint. Let me think.

Let me consider splitting $ a^4 $ into multiple terms so that I can apply AM-GM. For example, since the constraint involves $ ab^2 $, maybe I need to relate the exponents.

Alternatively, let me think of variables substitution. Let me set $ x = a $, $ y = b^2 $. Then the constraint becomes $ x y = 5 $, and we need to minimize $ x^4 + (y^2)^4 = x^4 + y^8 $. Wait, that's the same as before.

Alternatively, maybe express everything in terms of y. Since $ x = 5/y $, then the expression becomes $ (5/y)^4 + y^8 = 625 / y^4 + y^8 $. Wait, but earlier I had $ 625 / b^8 + b^8 $. Wait, maybe I made a substitution error here. Let me check:

Original substitution: $ y = b^2 $, so $ b = y^{1/2} $. Then $ b^8 = (y^{1/2})^8 = y^4 $. Therefore, the expression $ a^4 + b^8 = (5/y)^4 + y^4 = 625 / y^4 + y^4 $.

So, the function to minimize is $ f(y) = 625 / y^4 + y^4 $, where $ y > 0 $.

Wait, this seems different from before. Wait, earlier I had $ f(b) = 625 / b^8 + b^8 $. But here, with substitution $ y = b^2 $, it's $ 625 / y^4 + y^4 $. Which one is correct?

Let me check. Original expression: $ a^4 + b^8 $. With $ a = 5 / b^2 $, so $ a^4 = (5 / b^2)^4 = 625 / b^8 $. Then $ b^8 $ is just $ b^8 $. Therefore, total expression is $ 625 / b^8 + b^8 $.

But with substitution $ y = b^2 $, then $ b^8 = (b^2)^4 = y^4 $, and $ 625 / b^8 = 625 / y^4 $. Therefore, the expression is $ 625 / y^4 + y^4 $. So, same as before.

So, in this case, if I let $ y > 0 $, then the function is $ f(y) = 625 / y^4 + y^4 $.

So, maybe applying AM-GM here. Let me think.

The AM-GM inequality states that for non-negative real numbers, the arithmetic mean is greater than or equal to the geometric mean.

But here, we have two terms: $ 625 / y^4 $ and $ y^4 $. Let me consider them as two terms.

But AM-GM for two terms says that $ \frac{A + B}{2} \geq \sqrt{AB} $, so $ A + B \geq 2 \sqrt{AB} $.

So, applying that here:

$ \frac{625 / y^4 + y^4}{2} \geq \sqrt{ (625 / y^4) \times y^4 } = \sqrt{625} = 25 $

Therefore, multiplying both sides by 2:

$ 625 / y^4 + y^4 \geq 50 $

So, the minimum value is 50, achieved when $ 625 / y^4 = y^4 $, which implies $ y^8 = 625 $, so $ y = 625^{1/8} $.

Wait, but earlier, when I did substitution with $ b $, I found $ b = 5^{1/4} $, so $ y = b^2 = (5^{1/4})^2 = 5^{1/2} = \sqrt{5} $. Let me check if $ y = \sqrt{5} $ satisfies $ y^8 = 625 $:

Compute $ y^8 = (\sqrt{5})^8 = (5^{1/2})^8 = 5^{4} = 625 $. Yes! So that's correct. Therefore, equality holds when $ y = \sqrt{5} $, which gives the minimum value of 50.

Therefore, using AM-GM gives the same result as calculus. So, that confirms that the minimum value is indeed 50.

Alternatively, maybe I can use Hölder's inequality or other methods, but since both calculus and AM-GM give the same answer, I think that's solid.

But let me check if there's another way. For example, using Lagrange multipliers. Let me try that as another method.

Let me set up the Lagrangian. Let me denote the function to minimize as $ f(a, b) = a^4 + b^8 $, subject to the constraint $ g(a, b) = ab^2 - 5 = 0 $.

The Lagrangian is $ \mathcal{L}(a, b, \lambda) = a^4 + b^8 - \lambda(ab^2 - 5) $.

Taking partial derivatives:

Partial derivative with respect to $ a $:

$ \frac{\partial \mathcal{L}}{\partial a} = 4a^3 - \lambda b^2 = 0 $

Partial derivative with respect to $ b $:

$ \frac{\partial \mathcal{L}}{\partial b} = 8b^7 - \lambda \times 2ab = 0 $

Partial derivative with respect to $ \lambda $:

$ \frac{\partial \mathcal{L}}{\partial \lambda} = -(ab^2 - 5) = 0 $, which is the original constraint.

So, we have the system of equations:

1. $ 4a^3 = \lambda b^2 $
2. $ 8b^7 = 2ab \lambda $
3. $ ab^2 = 5 $

Let me solve equations 1 and 2 for $ \lambda $ and set them equal.

From equation 1: $ \lambda = \frac{4a^3}{b^2} $

From equation 2: $ 8b^7 = 2ab \lambda $, so divide both sides by 2ab:

$ \frac{8b^7}{2ab} = \lambda \implies \frac{4b^6}{a} = \lambda $

Therefore, equate the two expressions for $ \lambda $:

$ \frac{4a^3}{b^2} = \frac{4b^6}{a} $

Multiply both sides by $ b^2 $ and $ a $ to eliminate denominators:

$ 4a^3 \times a = 4b^6 \times b^2 $

Simplify:

Left side: $ 4a^4 $

Right side: $ 4b^8 $

Divide both sides by 4:

$ a^4 = b^8 $

Therefore, $ a^4 = b^8 \implies a = b^{2} $, since $ a $ and $ b $ are positive.

So, $ a = b^2 $. Now, substitute this into the constraint equation $ ab^2 = 5 $:

$ (b^2) \times b^2 = 5 \implies b^4 = 5 \implies b = 5^{1/4} $

Which is the same result as before. Then $ a = b^2 = (5^{1/4})^2 = 5^{1/2} = \sqrt{5} $.

Therefore, substituting back into $ a^4 + b^8 $:

$ (\sqrt{5})^4 + (5^{1/4})^8 = (5^{2}) + (5^{2}) = 25 + 25 = 50 $.

Same result again. So, regardless of the method, calculus, AM-GM, or Lagrange multipliers, I get the minimum value of 50.

Therefore, I can be confident that the minimum value is indeed 50.

But just to be thorough, let me check with some sample values. Suppose I pick a value of $ b $ different from $ 5^{1/4} $, compute $ a $, and check if $ a^4 + b^8 $ is larger than 50.

For example, let me take $ b = 1 $. Then $ ab^2 = 5 \implies a = 5 $. Then $ a^4 + b^8 = 5^4 + 1 = 625 + 1 = 626 $, which is way larger than 50.

Another example: take $ b = 5^{1/4} \approx 1.495 $. Let me compute $ b^8 $. Since $ b^4 = 5 $, so $ b^8 = (b^4)^2 = 25 $. Then $ a = \sqrt{5} \approx 2.236 $, so $ a^4 = (\sqrt{5})^4 = 25 $. So total is 50.

Another test: take $ b = 2 $. Then $ ab^2 = 5 \implies a = 5 / 4 $. Then $ a^4 = (5/4)^4 = 625 / 256 ≈ 2.441 $, and $ b^8 = 256 $. So total ≈ 2.441 + 256 ≈ 258.441, which is still larger than 50.

Another test: take $ b = 5^{1/4} \times k $, where $ k $ is some positive number. Let me see how the expression behaves.

Let me take $ k = 2 $, so $ b = 2 \times 5^{1/4} $. Then $ b^8 = (2^8) \times (5^{1/4})^8 = 256 \times 5^2 = 256 \times 25 = 6400 $. Then $ a = 5 / b^2 = 5 / (4 \times 5^{1/2}) ) = 5 / (4 \sqrt{5}) ) = \sqrt{5}/4 $. Then $ a^4 = ( \sqrt{5}/4 )^4 = (5^{1/2}/4)^4 = 5^2 / 4^4 = 25 / 256 ≈ 0.0977 $. Therefore, total expression ≈ 0.0977 + 6400 ≈ 6400.0977, which is way larger.

Alternatively, take $ k = 1/2 $, so $ b = (1/2) \times 5^{1/4} $. Then $ b^8 = (1/2)^8 \times 5^2 = 1/256 \times 25 = 25 / 256 ≈ 0.0977 $. Then $ a = 5 / b^2 = 5 / ( (1/2)^2 \times 5^{1/2} ) = 5 / ( (1/4) \times \sqrt{5} ) = 5 \times 4 / \sqrt{5} = 20 / \sqrt{5} = 4 \sqrt{5} ≈ 8.944 $. Then $ a^4 = (4 \sqrt{5})^4 = 4^4 \times (\sqrt{5})^4 = 256 \times 25 = 6400 $. Therefore, total expression ≈ 6400 + 0.0977 ≈ 6400.0977, again way larger.

So, this shows that when deviating from the critical point, the value increases, which confirms that 50 is indeed the minimum.

Alternatively, let me try a value closer to the critical point. Let me take $ b = 5^{1/4} \times 1.1 $. Let me compute $ b $:

First, $ 5^{1/4} ≈ 1.495 $. So, 1.1 times that is ≈ 1.6445. Then $ b^2 ≈ (1.6445)^2 ≈ 2.704 $. Then $ a = 5 / 2.704 ≈ 1.849 $. Then compute $ a^4 + b^8 $:

First, $ a^4 ≈ (1.849)^4 $. Let me compute step by step:

First, square 1.849: ≈ 3.418. Then square that: ≈ 11.685.

Then $ b^8 = (b^2)^4 ≈ (2.704)^4 $. Let me compute that:

First, square 2.704: ≈ 7.312. Then square that: ≈ 53.45.

Therefore, total expression ≈ 11.685 + 53.45 ≈ 65.135, which is still larger than 50.

So, even when moving slightly away from the critical point, the value increases.

Therefore, all these checks confirm that 50 is indeed the minimum.

Another thought: maybe there's a way to use Hölder's inequality? Let me recall Hölder's inequality.

Hölder's inequality generalizes the Cauchy-Schwarz inequality and relates the sum of products to the product of norms. However, I'm not sure if it's straightforward here. Let me think.

Alternatively, since we have exponents 4 and 8, maybe we can use weighted AM-GM.

Let me consider the terms in $ a^4 + b^8 $. Let me think of them as multiple terms so that the exponents can be balanced with the constraint.

The constraint is $ ab^2 = 5 $. Let me think of $ a^4 $ as $ a^4 $ and $ b^8 $ as $ b^8 $.

Alternatively, maybe split $ a^4 $ into multiple terms. For example, if I want to apply AM-GM with weights corresponding to the exponents in the constraint.

Alternatively, think of the exponents in the constraint and the expression to minimize.

Let me think of the constraint $ ab^2 = 5 $. Let me write the expression $ a^4 + b^8 $ as a combination of terms that can be related to the constraint.

Let me consider that $ a^4 = (a^1)^4 $, and $ b^8 = (b^2)^4 $. So, if I think of $ a^4 + b^8 = (a^1)^4 + (b^2)^4 $.

So, if I have two terms, each raised to the 4th power. Maybe I can apply AM-GM on these two terms.

But as before, AM-GM gives $ \frac{(a^1)^4 + (b^2)^4}{2} \geq \sqrt{(a^1)^4 (b^2)^4} = \sqrt{a^4 b^8} = a^2 b^4 $.

But then, how does that relate to the constraint $ ab^2 = 5 $?

Let me compute $ a^2 b^4 $. Since $ ab^2 = 5 $, then $ a^2 b^4 = a \times ab^2 \times b^2 = a \times 5 \times b^2 $. Wait, but that seems not helpful.

Alternatively, $ a^2 b^4 = (ab^2)^2 \times b^0 $? Wait, no. Let me compute:

Wait, $ a^2 b^4 = (ab^2)^2 \times b^0 $? No, $ (ab^2)^2 = a^2 b^4 $. So, $ a^2 b^4 = (ab^2)^2 = 5^2 = 25 $.

Oh! Therefore, from AM-GM:

$ \frac{a^4 + b^8}{2} \geq \sqrt{a^4 b^8} = a^2 b^4 = 25 $

Therefore, $ a^4 + b^8 \geq 50 $, with equality when $ a^4 = b^8 $, which is the same condition as before.

Therefore, this is another way to apply AM-GM directly, leading to the same result.

So, this shows that regardless of the method, the minimum is 50.

Therefore, I can be confident that the minimum value of $ a^4 + b^8 $ under the constraint $ ab^2 = 5 $ is 50.

Just to recap, the key steps were:

1. Use substitution to reduce the problem to a single variable.
2. Apply calculus to find critical points and confirm it's a minimum.
3. Use AM-GM inequality to find the minimum directly.
4. Verify with Lagrange multipliers.
5. Check with sample values to ensure the result is correct.

All methods converge to the same answer, so I think it's safe to conclude that the minimum value is indeed 50.

**Final Answer**
The minimum value of $ a^4 + b^8 $ is \boxed{50}.

To find the **minimum value** of the expression
$$
a^4 + b^8
$$
given the constraint
$$
ab^2 = 5
$$
with $ a, b > 0 $, we proceed as follows:

---

### Step 1: Use the Constraint to Eliminate One Variable

From the constraint $ ab^2 = 5 $, we can express $ a $ in terms of $ b $:
$$
a = \frac{5}{b^2}
$$

Substitute this into the expression $ a^4 + b^8 $:
$$
a^4 + b^8 = \left( \frac{5}{b^2} \right)^4 + b^8 = \frac{625}{b^8} + b^8
$$

So, the problem reduces to minimizing the function:
$$
f(b) = \frac{625}{b^8} + b^8
$$

---

### Step 2: Minimize the Function

Let’s define:
$$
f(b) = \frac{625}{b^8} + b^8
$$

Take the derivative with respect to $ b $:
$$
f'(b) = -\frac{5000}{b^9} + 8b^7
$$

Set $ f'(b) = 0 $ to find critical points:
$$
-\frac{5000}{b^9} + 8b^7 = 0 \Rightarrow 8b^7 = \frac{5000}{b^9}
$$

Multiply both sides by $ b^9 $:
$$
8b^{16} = 5000 \Rightarrow b^{16} = 625
$$

Take the 16th root:
$$
b = 625^{1/16} = (5^4)^{1/16} = 5^{1/4}
$$

Now compute $ a $ using the constraint:
$$
a = \frac{5}{b^2} = \frac{5}{(5^{1/4})^2} = \frac{5}{5^{1/2}} = 5^{1/2} = \sqrt{5}
$$

---

### Step 3: Evaluate the Expression at the Critical Point

Now compute:
$$
a^4 + b^8 = (\sqrt{5})^4 + (5^{1/4})^8 = 5^2 + 5^2 = 25 + 25 = 50
$$

---

### Step 4: Confirm It's a Minimum

We can confirm this is a minimum by checking the second derivative:
$$
f''(b) = \frac{45000}{b^{10}} + 56b^6
$$
Since $ b > 0 $, both terms are positive, so $ f''(b) > 0 $, confirming a local minimum.

Also, as $ b \to 0^+ $ or $ b \to \infty $, the function $ f(b) \to \infty $, so the critical point is the **global minimum**.

---

### Step 5: Alternative Verification via AM-GM Inequality

We can also apply the **AM-GM inequality** to the expression $ a^4 + b^8 $, treating it as two terms:

$$
\frac{a^4 + b^8}{2} \geq \sqrt{a^4 b^8} = a^2 b^4
$$

From the constraint $ ab^2 = 5 $, we have:
$$
a^2 b^4 = (ab^2)^2 = 5^2 = 25
$$

So:
$$
\frac{a^4 + b^8}{2} \geq 25 \Rightarrow a^4 + b^8 \geq 50
$$

Equality holds when $ a^4 = b^8 $, which is consistent with the earlier solution.

---

### Final Answer

$$
\boxed{50}
$$

[Assistant]
Explanation:
Okay, so I need to find the minimum value of $ a^4 + b^8 $ given that $ ab^2 = 5 $ where $ a $ and $ b $ are positive real numbers. Hmm, let me think about how to approach this.

First, since there's a constraint $ ab^2 = 5 $, maybe I can use substitution to express one variable in terms of the other and then minimize the function. Let me try that.

Let me solve the constraint equation for $ a $. If $ ab^2 = 5 $, then $ a = \frac{5}{b^2} $. Now, substitute this into the expression we need to minimize, which is $ a^4 + b^8 $.

So substituting $ a $, we get:

$ \left( \frac{5}{b^2} \right)^4 + b^8 $

Let me compute that:

First, $ \left( \frac{5}{b^2} \right)^4 = \frac{5^4}{b^8} = \frac{625}{b^8} $. Therefore, the expression becomes:

$ \frac{625}{b^8} + b^8 $

So now, the problem reduces to finding the minimum of the function $ f(b) = \frac{625}{b^8} + b^8 $ where $ b > 0 $.

Hmm, okay. To find the minimum, I can take the derivative of $ f(b) $ with respect to $ b $, set it equal to zero, and solve for $ b $. Then check if it's a minimum using the second derivative or some other method. Let me try that.

First, compute the derivative $ f'(b) $:

$ f(b) = 625 b^{-8} + b^8 $

So, derivative term by term:

The derivative of $ 625 b^{-8} $ is $ 625 \times (-8) b^{-9} = -5000 b^{-9} $

The derivative of $ b^8 $ is $ 8 b^7 $

Therefore, $ f'(b) = -5000 b^{-9} + 8 b^7 $

Set this equal to zero for critical points:

$ -5000 b^{-9} + 8 b^7 = 0 $

Let me rewrite this equation:

$ 8 b^7 = 5000 b^{-9} $

Multiply both sides by $ b^9 $ to eliminate the negative exponent:

$ 8 b^{16} = 5000 $

Then, divide both sides by 8:

$ b^{16} = \frac{5000}{8} $

Calculate $ 5000 / 8 $:

Well, 5000 divided by 8 is 625. So:

$ b^{16} = 625 $

Now, solve for $ b $. Since $ b > 0 $, we can take the 16th root:

$ b = 625^{1/16} $

Hmm, let me simplify 625. 625 is 5^4, right? Because 5^4 = 625. So:

$ b = (5^4)^{1/16} = 5^{4/16} = 5^{1/4} $

So $ b = 5^{1/4} $. Let me check that again.

Yes, because $ 5^{1/4} $ is the fourth root of 5, which is the same as $ 5^{0.25} $.

Alternatively, since $ b^{16} = 5^4 $, taking both sides to the power of 1/16 gives $ b = 5^{4/16} = 5^{1/4} $. Correct.

So, now that I have $ b $, I can find $ a $ using the original constraint $ ab^2 = 5 $. Let me compute $ a $:

$ a = \frac{5}{b^2} $

Since $ b = 5^{1/4} $, then $ b^2 = (5^{1/4})^2 = 5^{1/2} = \sqrt{5} $. Therefore:

$ a = \frac{5}{\sqrt{5}} = \sqrt{5} $. Because $ 5 / \sqrt{5} = \sqrt{5} \times \sqrt{5} / \sqrt{5} = \sqrt{5} $.

So, $ a = \sqrt{5} $, $ b = 5^{1/4} $.

Now, I need to compute the value of $ a^4 + b^8 $ at these values to find the minimum. Let me compute each term:

First, $ a^4 $:

Since $ a = \sqrt{5} = 5^{1/2} $, then $ a^4 = (5^{1/2})^4 = 5^{2} = 25 $.

Next, $ b^8 $:

Since $ b = 5^{1/4} $, then $ b^8 = (5^{1/4})^8 = 5^{2} = 25 $.

Therefore, $ a^4 + b^8 = 25 + 25 = 50 $.

So, the minimum value is 50? Wait, but let me check if this is indeed a minimum. Maybe I should verify using the second derivative or check the behavior of the function.

Alternatively, since we found a critical point and the function tends to infinity as $ b $ approaches 0 or infinity, this critical point is likely to be the minimum. Let me confirm with the second derivative.

Compute the second derivative $ f''(b) $:

We had $ f'(b) = -5000 b^{-9} + 8 b^7 $

Differentiate again:

The derivative of $ -5000 b^{-9} $ is $ -5000 \times (-9) b^{-10} = 45000 b^{-10} $

The derivative of $ 8 b^7 $ is $ 56 b^6 $

Therefore, $ f''(b) = 45000 b^{-10} + 56 b^6 $

Since $ b > 0 $, both terms are positive. Therefore, $ f''(b) > 0 $, which means the function is convex at this critical point, so it is indeed a local minimum. Since the function tends to infinity as $ b $ approaches 0 or infinity, this local minimum is the global minimum.

Therefore, the minimum value of $ a^4 + b^8 $ is 50.

But let me think again if there's another way to approach this problem, maybe using inequalities like AM-GM? Maybe that could be another method to confirm the answer.

Let me try using the AM-GM inequality. The expression we need to minimize is $ a^4 + b^8 $. However, the constraint is $ ab^2 = 5 $.

But AM-GM usually applies to sums of terms with exponents that can be balanced. Let me see.

Alternatively, maybe I can write $ a^4 + b^8 $ as a sum of terms that can be related to the constraint. Let me think.

Let me consider splitting $ a^4 $ into multiple terms so that I can apply AM-GM. For example, since the constraint involves $ ab^2 $, maybe I need to relate the exponents.

Alternatively, let me think of variables substitution. Let me set $ x = a $, $ y = b^2 $. Then the constraint becomes $ x y = 5 $, and we need to minimize $ x^4 + (y^2)^4 = x^4 + y^8 $. Wait, that's the same as before.

Alternatively, maybe express everything in terms of y. Since $ x = 5/y $, then the expression becomes $ (5/y)^4 + y^8 = 625 / y^4 + y^8 $. Wait, but earlier I had $ 625 / b^8 + b^8 $. Wait, maybe I made a substitution error here. Let me check:

Original substitution: $ y = b^2 $, so $ b = y^{1/2} $. Then $ b^8 = (y^{1/2})^8 = y^4 $. Therefore, the expression $ a^4 + b^8 = (5/y)^4 + y^4 = 625 / y^4 + y^4 $.

So, the function to minimize is $ f(y) = 625 / y^4 + y^4 $, where $ y > 0 $.

Wait, this seems different from before. Wait, earlier I had $ f(b) = 625 / b^8 + b^8 $. But here, with substitution $ y = b^2 $, it's $ 625 / y^4 + y^4 $. Which one is correct?

Let me check. Original expression: $ a^4 + b^8 $. With $ a = 5 / b^2 $, so $ a^4 = (5 / b^2)^4 = 625 / b^8 $. Then $ b^8 $ is just $ b^8 $. Therefore, total expression is $ 625 / b^8 + b^8 $.

But with substitution $ y = b^2 $, then $ b^8 = (b^2)^4 = y^4 $, and $ 625 / b^8 = 625 / y^4 $. Therefore, the expression is $ 625 / y^4 + y^4 $. So, same as before.

So, in this case, if I let $ y > 0 $, then the function is $ f(y) = 625 / y^4 + y^4 $.

So, maybe applying AM-GM here. Let me think.

The AM-GM inequality states that for non-negative real numbers, the arithmetic mean is greater than or equal to the geometric mean.

But here, we have two terms: $ 625 / y^4 $ and $ y^4 $. Let me consider them as two terms.

But AM-GM for two terms says that $ \frac{A + B}{2} \geq \sqrt{AB} $, so $ A + B \geq 2 \sqrt{AB} $.

So, applying that here:

$ \frac{625 / y^4 + y^4}{2} \geq \sqrt{ (625 / y^4) \times y^4 } = \sqrt{625} = 25 $

Therefore, multiplying both sides by 2:

$ 625 / y^4 + y^4 \geq 50 $

So, the minimum value is 50, achieved when $ 625 / y^4 = y^4 $, which implies $ y^8 = 625 $, so $ y = 625^{1/8} $.

Wait, but earlier, when I did substitution with $ b $, I found $ b = 5^{1/4} $, so $ y = b^2 = (5^{1/4})^2 = 5^{1/2} = \sqrt{5} $. Let me check if $ y = \sqrt{5} $ satisfies $ y^8 = 625 $:

Compute $ y^8 = (\sqrt{5})^8 = (5^{1/2})^8 = 5^{4} = 625 $. Yes! So that's correct. Therefore, equality holds when $ y = \sqrt{5} $, which gives the minimum value of 50.

Therefore, using AM-GM gives the same result as calculus. So, that confirms that the minimum value is indeed 50.

Alternatively, maybe I can use Hölder's inequality or other methods, but since both calculus and AM-GM give the same answer, I think that's solid.

But let me check if there's another way. For example, using Lagrange multipliers. Let me try that as another method.

Let me set up the Lagrangian. Let me denote the function to minimize as $ f(a, b) = a^4 + b^8 $, subject to the constraint $ g(a, b) = ab^2 - 5 = 0 $.

The Lagrangian is $ \mathcal{L}(a, b, \lambda) = a^4 + b^8 - \lambda(ab^2 - 5) $.

Taking partial derivatives:

Partial derivative with respect to $ a $:

$ \frac{\partial \mathcal{L}}{\partial a} = 4a^3 - \lambda b^2 = 0 $

Partial derivative with respect to $ b $:

$ \frac{\partial \mathcal{L}}{\partial b} = 8b^7 - \lambda \times 2ab = 0 $

Partial derivative with respect to $ \lambda $:

$ \frac{\partial \mathcal{L}}{\partial \lambda} = -(ab^2 - 5) = 0 $, which is the original constraint.

So, we have the system of equations:

1. $ 4a^3 = \lambda b^2 $
2. $ 8b^7 = 2ab \lambda $
3. $ ab^2 = 5 $

Let me solve equations 1 and 2 for $ \lambda $ and set them equal.

From equation 1: $ \lambda = \frac{4a^3}{b^2} $

From equation 2: $ 8b^7 = 2ab \lambda $, so divide both sides by 2ab:

$ \frac{8b^7}{2ab} = \lambda \implies \frac{4b^6}{a} = \lambda $

Therefore, equate the two expressions for $ \lambda $:

$ \frac{4a^3}{b^2} = \frac{4b^6}{a} $

Multiply both sides by $ b^2 $ and $ a $ to eliminate denominators:

$ 4a^3 \times a = 4b^6 \times b^2 $

Simplify:

Left side: $ 4a^4 $

Right side: $ 4b^8 $

Divide both sides by 4:

$ a^4 = b^8 $

Therefore, $ a^4 = b^8 \implies a = b^{2} $, since $ a $ and $ b $ are positive.

So, $ a = b^2 $. Now, substitute this into the constraint equation $ ab^2 = 5 $:

$ (b^2) \times b^2 = 5 \implies b^4 = 5 \implies b = 5^{1/4} $

Which is the same result as before. Then $ a = b^2 = (5^{1/4})^2 = 5^{1/2} = \sqrt{5} $.

Therefore, substituting back into $ a^4 + b^8 $:

$ (\sqrt{5})^4 + (5^{1/4})^8 = (5^{2}) + (5^{2}) = 25 + 25 = 50 $.

Same result again. So, regardless of the method, calculus, AM-GM, or Lagrange multipliers, I get the minimum value of 50.

Therefore, I can be confident that the minimum value is indeed 50.

But just to be thorough, let me check with some sample values. Suppose I pick a value of $ b $ different from $ 5^{1/4} $, compute $ a $, and check if $ a^4 + b^8 $ is larger than 50.

For example, let me take $ b = 1 $. Then $ ab^2 = 5 \implies a = 5 $. Then $ a^4 + b^8 = 5^4 + 1 = 625 + 1 = 626 $, which is way larger than 50.

Another example: take $ b = 5^{1/4} \approx 1.495 $. Let me compute $ b^8 $. Since $ b^4 = 5 $, so $ b^8 = (b^4)^2 = 25 $. Then $ a = \sqrt{5} \approx 2.236 $, so $ a^4 = (\sqrt{5})^4 = 25 $. So total is 50.

Another test: take $ b = 2 $. Then $ ab^2 = 5 \implies a = 5 / 4 $. Then $ a^4 = (5/4)^4 = 625 / 256 ≈ 2.441 $, and $ b^8 = 256 $. So total ≈ 2.441 + 256 ≈ 258.441, which is still larger than 50.

Another test: take $ b = 5^{1/4} \times k $, where $ k $ is some positive number. Let me see how the expression behaves.

Let me take $ k = 2 $, so $ b = 2 \times 5^{1/4} $. Then $ b^8 = (2^8) \times (5^{1/4})^8 = 256 \times 5^2 = 256 \times 25 = 6400 $. Then $ a = 5 / b^2 = 5 / (4 \times 5^{1/2}) ) = 5 / (4 \sqrt{5}) ) = \sqrt{5}/4 $. Then $ a^4 = ( \sqrt{5}/4 )^4 = (5^{1/2}/4)^4 = 5^2 / 4^4 = 25 / 256 ≈ 0.0977 $. Therefore, total expression ≈ 0.0977 + 6400 ≈ 6400.0977, which is way larger.

Alternatively, take $ k = 1/2 $, so $ b = (1/2) \times 5^{1/4} $. Then $ b^8 = (1/2)^8 \times 5^2 = 1/256 \times 25 = 25 / 256 ≈ 0.0977 $. Then $ a = 5 / b^2 = 5 / ( (1/2)^2 \times 5^{1/2} ) = 5 / ( (1/4) \times \sqrt{5} ) = 5 \times 4 / \sqrt{5} = 20 / \sqrt{5} = 4 \sqrt{5} ≈ 8.944 $. Then $ a^4 = (4 \sqrt{5})^4 = 4^4 \times (\sqrt{5})^4 = 256 \times 25 = 6400 $. Therefore, total expression ≈ 6400 + 0.0977 ≈ 6400.0977, again way larger.

So, this shows that when deviating from the critical point, the value increases, which confirms that 50 is indeed the minimum.

Alternatively, let me try a value closer to the critical point. Let me take $ b = 5^{1/4} \times 1.1 $. Let me compute $ b $:

First, $ 5^{1/4} ≈ 1.495 $. So, 1.1 times that is ≈ 1.6445. Then $ b^2 ≈ (1.6445)^2 ≈ 2.704 $. Then $ a = 5 / 2.704 ≈ 1.849 $. Then compute $ a^4 + b^8 $:

First, $ a^4 ≈ (1.849)^4 $. Let me compute step by step:

First, square 1.849: ≈ 3.418. Then square that: ≈ 11.685.

Then $ b^8 = (b^2)^4 ≈ (2.704)^4 $. Let me compute that:

First, square 2.704: ≈ 7.312. Then square that: ≈ 53.45.

Therefore, total expression ≈ 11.685 + 53.45 ≈ 65.135, which is still larger than 50.

So, even when moving slightly away from the critical point, the value increases.

Therefore, all these checks confirm that 50 is indeed the minimum.

Another thought: maybe there's a way to use Hölder's inequality? Let me recall Hölder's inequality.

Hölder's inequality generalizes the Cauchy-Schwarz inequality and relates the sum of products to the product of norms. However, I'm not sure if it's straightforward here. Let me think.

Alternatively, since we have exponents 4 and 8, maybe we can use weighted AM-GM.

Let me consider the terms in $ a^4 + b^8 $. Let me think of them as multiple terms so that the exponents can be balanced with the constraint.

The constraint is $ ab^2 = 5 $. Let me think of $ a^4 $ as $ a^4 $ and $ b^8 $ as $ b^8 $.

Alternatively, maybe split $ a^4 $ into multiple terms. For example, if I want to apply AM-GM with weights corresponding to the exponents in the constraint.

Alternatively, think of the exponents in the constraint and the expression to minimize.

Let me think of the constraint $ ab^2 = 5 $. Let me write the expression $ a^4 + b^8 $ as a combination of terms that can be related to the constraint.

Let me consider that $ a^4 = (a^1)^4 $, and $ b^8 = (b^2)^4 $. So, if I think of $ a^4 + b^8 = (a^1)^4 + (b^2)^4 $.

So, if I have two terms, each raised to the 4th power. Maybe I can apply AM-GM on these two terms.

But as before, AM-GM gives $ \frac{(a^1)^4 + (b^2)^4}{2} \geq \sqrt{(a^1)^4 (b^2)^4} = \sqrt{a^4 b^8} = a^2 b^4 $.

But then, how does that relate to the constraint $ ab^2 = 5 $?

Let me compute $ a^2 b^4 $. Since $ ab^2 = 5 $, then $ a^2 b^4 = a \times ab^2 \times b^2 = a \times 5 \times b^2 $. Wait, but that seems not helpful.

Alternatively, $ a^2 b^4 = (ab^2)^2 \times b^0 $? Wait, no. Let me compute:

Wait, $ a^2 b^4 = (ab^2)^2 \times b^0 $? No, $ (ab^2)^2 = a^2 b^4 $. So, $ a^2 b^4 = (ab^2)^2 = 5^2 = 25 $.

Oh! Therefore, from AM-GM:

$ \frac{a^4 + b^8}{2} \geq \sqrt{a^4 b^8} = a^2 b^4 = 25 $

Therefore, $ a^4 + b^8 \geq 50 $, with equality when $ a^4 = b^8 $, which is the same condition as before.

Therefore, this is another way to apply AM-GM directly, leading to the same result.

So, this shows that regardless of the method, the minimum is 50.

Therefore, I can be confident that the minimum value of $ a^4 + b^8 $ under the constraint $ ab^2 = 5 $ is 50.

Just to recap, the key steps were:

1. Use substitution to reduce the problem to a single variable.
2. Apply calculus to find critical points and confirm it's a minimum.
3. Use AM-GM inequality to find the minimum directly.
4. Verify with Lagrange multipliers.
5. Check with sample values to ensure the result is correct.

All methods converge to the same answer, so I think it's safe to conclude that the minimum value is indeed 50.

**Final Answer**
The minimum value of $ a^4 + b^8 $ is \boxed{50}.

To find the **minimum value** of the expression
$$
a^4 + b^8
$$
given the constraint
$$
ab^2 = 5
$$
with $ a, b > 0 $, we proceed as follows:

---

### Step 1: Use the Constraint to Eliminate One Variable

From the constraint $ ab^2 = 5 $, we can express $ a $ in terms of $ b $:
$$
a = \frac{5}{b^2}
$$

Substitute this into the expression $ a^4 + b^8 $:
$$
a^4 + b^8 = \left( \frac{5}{b^2} \right)^4 + b^8 = \frac{625}{b^8} + b^8
$$

So, the problem reduces to minimizing the function:
$$
f(b) = \frac{625}{b^8} + b^8
$$

---

### Step 2: Minimize the Function

Let’s define:
$$
f(b) = \frac{625}{b^8} + b^8
$$

Take the derivative with respect to $ b $:
$$
f'(b) = -\frac{5000}{b^9} + 8b^7
$$

Set $ f'(b) = 0 $ to find critical points:
$$
-\frac{5000}{b^9} + 8b^7 = 0 \Rightarrow 8b^7 = \frac{5000}{b^9}
$$

Multiply both sides by $ b^9 $:
$$
8b^{16} = 5000 \Rightarrow b^{16} = 625
$$

Take the 16th root:
$$
b = 625^{1/16} = (5^4)^{1/16} = 5^{1/4}
$$

Now compute $ a $ using the constraint:
$$
a = \frac{5}{b^2} = \frac{5}{(5^{1/4})^2} = \frac{5}{5^{1/2}} = 5^{1/2} = \sqrt{5}
$$

---

### Step 3: Evaluate the Expression at the Critical Point

Now compute:
$$
a^4 + b^8 = (\sqrt{5})^4 + (5^{1/4})^8 = 5^2 + 5^2 = 25 + 25 = 50
$$

---

### Step 4: Confirm It's a Minimum

We can confirm this is a minimum by checking the second derivative:
$$
f''(b) = \frac{45000}{b^{10}} + 56b^6
$$
Since $ b > 0 $, both terms are positive, so $ f''(b) > 0 $, confirming a local minimum.

Also, as $ b \to 0^+ $ or $ b \to \infty $, the function $ f(b) \to \infty $, so the critical point is the **global minimum**.

---

### Step 5: Alternative Verification via AM-GM Inequality

We can also apply the **AM-GM inequality** to the expression $ a^4 + b^8 $, treating it as two terms:

$$
\frac{a^4 + b^8}{2} \geq \sqrt{a^4 b^8} = a^2 b^4
$$

From the constraint $ ab^2 = 5 $, we have:
$$
a^2 b^4 = (ab^2)^2 = 5^2 = 25
$$

So:
$$
\frac{a^4 + b^8}{2} \geq 25 \Rightarrow a^4 + b^8 \geq 50
$$

Equality holds when $ a^4 = b^8 $, which is consistent with the earlier solution.

---

### Final Answer

$$
\boxed{50}
$$

Solution:
By AM-GM,
\[a^4 + b^8 \ge 2 \sqrt{a^4 b^8} = 2a^2 b^4 = 2(ab^2)^2 = 50.\]Equality occurs when $a^4 = b^8$ and $ab^2 = 5;$ we can solve to get $a = \sqrt{5}$ and $b = \sqrt[4]{5},$ so the minimum value is $\boxed{50}.$


Explanation:
Okay, so I need to figure out the value of the square symbol, which is represented by □. The problem says that the symbols △, □, ♦, and ♣ represent four different integers from 1 to 9. There are three equations given:

1. △ + □ = ♣
2. △ + △ = ♦ + ♦ + ♦ + ♦ + ♦
3. △ + △ = ♣ + ♦

Alright, let me start by writing down the equations more clearly with variables to make it easier. Let me assign letters to each symbol:

Let’s say:
- Let a = △
- Let b = □
- Let c = ♦
- Let d = ♣

So the equations become:

1. a + b = d
2. 2a = 5c
3. 2a = d + c

So now we have three equations with four variables, but since all variables are integers from 1 to 9 and they are all different, we can try to solve this system.

First, let me note that equation 2 and equation 3 both equal 2a. Therefore, they must equal each other. So:

From equation 2: 2a = 5c

From equation 3: 2a = d + c

Therefore, 5c = d + c. Let me solve for d:

Subtract c from both sides: 5c - c = d => 4c = d. Therefore, d = 4c.

So from equation 1: a + b = d = 4c. Therefore, a + b = 4c.

But also from equation 2: 2a = 5c. Therefore, we can express a in terms of c: a = (5c)/2.

Since a must be an integer (as all symbols represent integers from 1 to 9), 5c must be even. Therefore, c must be even because 5 is odd. So c is even. Also, since c is between 1 and 9, possible values for c are 2, 4, 6, 8.

But let's check each possible value of c and see which ones work.

First, let's note that a = (5c)/2. Since a must be an integer between 1 and 9, let's check possible c:

Case 1: c = 2

Then a = (5*2)/2 = 5. So a = 5. Then from equation 3: 2a = d + c => 10 = d + 2 => d = 8. So d = 8. Then from equation 1: a + b = d => 5 + b = 8 => b = 3. So b = 3. Now check if all numbers are distinct: a=5, b=3, c=2, d=8. All different. So that's possible. Let me check if they are all between 1-9: yes. So that's a possible solution.

Case 2: c = 4

Then a = (5*4)/2 = 10. But a must be between 1-9. 10 is invalid. So this case is impossible.

Case 3: c = 6

Then a = (5*6)/2 = 15. Again, too big. 15 is outside 1-9. Not valid.

Case 4: c = 8

a = (5*8)/2 = 20. Also too big. So invalid.

Therefore, only c=2 gives a valid value for a. Therefore, c=2, a=5, d=8, b=3. Therefore, the symbols are:

△ = a = 5

□ = b = 3

♦ = c = 2

♣ = d = 8

But let me check if all numbers are different. 5, 3, 2, 8. Yes, all different. And they are all between 1-9. So that seems to satisfy all equations.

Let me verify each equation:

First equation: △ + □ = 5 + 3 = 8, which is ♣ = 8. Correct.

Second equation: △ + △ = 5 + 5 = 10. On the other side, ♦ + ♦ + ♦ + ♦ + ♦ = 2 + 2 + 2 + 2 + 2 = 10. Correct.

Third equation: △ + △ = 10. On the other side, ♣ + ♦ = 8 + 2 = 10. Correct.

So all equations are satisfied. Therefore, the value of □ is 3. Therefore, the answer is 3. Wait, but let me check if there are other possibilities. Wait, earlier when c=2, we had a=5, d=8, b=3. Are there any other possible values?

Wait, earlier when I considered c must be even, so c=2,4,6,8. But only c=2 gives a valid a. So that's the only solution. Therefore, the answer is 3. So □ is 3. Therefore, the answer is \boxed{3}.

But let me think again. Wait, is there a possibility that c is not even? Wait, earlier I said that since 5c must be even, so c must be even. Because 5 is odd. So odd times c is even. Therefore, c must be even. Therefore, only even numbers. So that's correct. Therefore, only c=2 is possible. Therefore, the answer is 3. So I think that's correct.

But let me check again. Let me think if there's another way. Suppose someone might think that maybe there's another solution. Let me check again.

Suppose c is even. So c=2,4,6,8. For c=2, a=5, which is okay. For c=4, a=10, which is invalid. For c=6, a=15, invalid. For c=8, a=20, invalid. So no other possibilities. Therefore, only solution is c=2, a=5, d=8, b=3. Therefore, □ is 3. So answer is 3. Therefore, I think that's correct.

But let me check if there's another way to approach this problem. Maybe by substitution.

From equation 2: 2a = 5c. Therefore, 5 divides 2a. Since 5 is prime, and 5 doesn't divide 2, so 5 divides a. Therefore, a must be a multiple of 5. Since a is between 1-9, possible values for a are 5. Because 5*1=5, 5*2=10 which is too big. Therefore, a=5. Therefore, from equation 2: 2*5=5c => 10=5c => c=2. Therefore, c=2. Then from equation 3: 2a = d + c => 10 = d + 2 => d=8. Then from equation 1: a + b = d => 5 + b = 8 => b=3. Therefore, same result. Therefore, this confirms that a=5, c=2, d=8, b=3. Therefore, □ is 3. Therefore, answer is 3. So this seems solid.

Therefore, I think the answer is 3.

**Final Answer**
\boxed{3}

To solve the problem, we are given four symbols:
- △, □, ♦, and ♣, which represent **four different integers from 1 to 9**.
- Three equations involving these symbols:

$$
\begin{align*}
(1)\quad \triangle + \square &= \clubsuit \\
(2)\quad \triangle + \triangle &= \diamond + \diamond + \diamond + \diamond + \diamond \\
(3)\quad \triangle + \triangle &= \clubsuit + \diamond
\end{align*}
$$

---

### Step 1: Assign Variables

Let’s assign variables to the symbols for clarity:

- Let $ a = \triangle $
- Let $ b = \square $
- Let $ c = \diamond $
- Let $ d = \clubsuit $

Now rewrite the equations:

$$
\begin{align*}
(1)\quad a + b &= d \\
(2)\quad 2a &= 5c \\
(3)\quad 2a &= d + c
\end{align*}
$$

---

### Step 2: Use Equations (2) and (3) to Eliminate $ d $

From equations (2) and (3), both equal $ 2a $, so we can equate them:

$$
5c = d + c \Rightarrow 4c = d
$$

So, $ d = 4c $

---

### Step 3: Substitute into Equation (1)

From equation (1):
$$
a + b = d = 4c
$$

So,
$$
a + b = 4c \tag{4}
$$

From equation (2):
$$
2a = 5c \Rightarrow a = \frac{5c}{2}
$$

Since $ a $ must be an integer, $ 5c $ must be even. Since 5 is odd, $ c $ must be even.

---

### Step 4: Try Possible Values for $ c $

We know $ c \in \{1, 2, 3, ..., 9\} $, and $ c $ must be even. So possible values for $ c $ are:
$$
c = 2, 4, 6, 8
$$

Try each:

#### Case 1: $ c = 2 $

- $ a = \frac{5 \cdot 2}{2} = 5 $
- $ d = 4c = 8 $
- From equation (1): $ a + b = d \Rightarrow 5 + b = 8 \Rightarrow b = 3 $

So we have:
- $ a = 5 $
- $ b = 3 $
- $ c = 2 $
- $ d = 8 $

Check if all values are distinct and between 1 and 9:
✅ All values are distinct and in the range.

Now verify all equations:

- (1): $ a + b = 5 + 3 = 8 = d $ ✅
- (2): $ 2a = 10 $, and $ 5c = 10 $ ✅
- (3): $ 2a = 10 $, and $ d + c = 8 + 2 = 10 $ ✅

All equations are satisfied.

---

### Step 5: Check for Other Possible Values of $ c $

Try $ c = 4 $:
- $ a = \frac{5 \cdot 4}{2} = 10 $ → Invalid (not in 1–9)

Try $ c = 6 $:
- $ a = \frac{5 \cdot 6}{2} = 15 $ → Invalid

Try $ c = 8 $:
- $ a = \frac{5 \cdot 8}{2} = 20 $ → Invalid

So, only $ c = 2 $ gives valid values for all variables.

---

### Final Answer

From the above, the value of $ \square $ is:

$$
\boxed{3}
$$

[Assistant]
Explanation:
Okay, so I need to figure out the value of the square symbol, which is represented by □. The problem says that the symbols △, □, ♦, and ♣ represent four different integers from 1 to 9. There are three equations given:

1. △ + □ = ♣
2. △ + △ = ♦ + ♦ + ♦ + ♦ + ♦
3. △ + △ = ♣ + ♦

Alright, let me start by writing down the equations more clearly with variables to make it easier. Let me assign letters to each symbol:

Let’s say:
- Let a = △
- Let b = □
- Let c = ♦
- Let d = ♣

So the equations become:

1. a + b = d
2. 2a = 5c
3. 2a = d + c

So now we have three equations with four variables, but since all variables are integers from 1 to 9 and they are all different, we can try to solve this system.

First, let me note that equation 2 and equation 3 both equal 2a. Therefore, they must equal each other. So:

From equation 2: 2a = 5c

From equation 3: 2a = d + c

Therefore, 5c = d + c. Let me solve for d:

Subtract c from both sides: 5c - c = d => 4c = d. Therefore, d = 4c.

So from equation 1: a + b = d = 4c. Therefore, a + b = 4c.

But also from equation 2: 2a = 5c. Therefore, we can express a in terms of c: a = (5c)/2.

Since a must be an integer (as all symbols represent integers from 1 to 9), 5c must be even. Therefore, c must be even because 5 is odd. So c is even. Also, since c is between 1 and 9, possible values for c are 2, 4, 6, 8.

But let's check each possible value of c and see which ones work.

First, let's note that a = (5c)/2. Since a must be an integer between 1 and 9, let's check possible c:

Case 1: c = 2

Then a = (5*2)/2 = 5. So a = 5. Then from equation 3: 2a = d + c => 10 = d + 2 => d = 8. So d = 8. Then from equation 1: a + b = d => 5 + b = 8 => b = 3. So b = 3. Now check if all numbers are distinct: a=5, b=3, c=2, d=8. All different. So that's possible. Let me check if they are all between 1-9: yes. So that's a possible solution.

Case 2: c = 4

Then a = (5*4)/2 = 10. But a must be between 1-9. 10 is invalid. So this case is impossible.

Case 3: c = 6

Then a = (5*6)/2 = 15. Again, too big. 15 is outside 1-9. Not valid.

Case 4: c = 8

a = (5*8)/2 = 20. Also too big. So invalid.

Therefore, only c=2 gives a valid value for a. Therefore, c=2, a=5, d=8, b=3. Therefore, the symbols are:

△ = a = 5

□ = b = 3

♦ = c = 2

♣ = d = 8

But let me check if all numbers are different. 5, 3, 2, 8. Yes, all different. And they are all between 1-9. So that seems to satisfy all equations.

Let me verify each equation:

First equation: △ + □ = 5 + 3 = 8, which is ♣ = 8. Correct.

Second equation: △ + △ = 5 + 5 = 10. On the other side, ♦ + ♦ + ♦ + ♦ + ♦ = 2 + 2 + 2 + 2 + 2 = 10. Correct.

Third equation: △ + △ = 10. On the other side, ♣ + ♦ = 8 + 2 = 10. Correct.

So all equations are satisfied. Therefore, the value of □ is 3. Therefore, the answer is 3. Wait, but let me check if there are other possibilities. Wait, earlier when c=2, we had a=5, d=8, b=3. Are there any other possible values?

Wait, earlier when I considered c must be even, so c=2,4,6,8. But only c=2 gives a valid a. So that's the only solution. Therefore, the answer is 3. So □ is 3. Therefore, the answer is \boxed{3}.

But let me think again. Wait, is there a possibility that c is not even? Wait, earlier I said that since 5c must be even, so c must be even. Because 5 is odd. So odd times c is even. Therefore, c must be even. Therefore, only even numbers. So that's correct. Therefore, only c=2 is possible. Therefore, the answer is 3. So I think that's correct.

But let me check again. Let me think if there's another way. Suppose someone might think that maybe there's another solution. Let me check again.

Suppose c is even. So c=2,4,6,8. For c=2, a=5, which is okay. For c=4, a=10, which is invalid. For c=6, a=15, invalid. For c=8, a=20, invalid. So no other possibilities. Therefore, only solution is c=2, a=5, d=8, b=3. Therefore, □ is 3. So answer is 3. Therefore, I think that's correct.

But let me check if there's another way to approach this problem. Maybe by substitution.

From equation 2: 2a = 5c. Therefore, 5 divides 2a. Since 5 is prime, and 5 doesn't divide 2, so 5 divides a. Therefore, a must be a multiple of 5. Since a is between 1-9, possible values for a are 5. Because 5*1=5, 5*2=10 which is too big. Therefore, a=5. Therefore, from equation 2: 2*5=5c => 10=5c => c=2. Therefore, c=2. Then from equation 3: 2a = d + c => 10 = d + 2 => d=8. Then from equation 1: a + b = d => 5 + b = 8 => b=3. Therefore, same result. Therefore, this confirms that a=5, c=2, d=8, b=3. Therefore, □ is 3. Therefore, answer is 3. So this seems solid.

Therefore, I think the answer is 3.

**Final Answer**
\boxed{3}

To solve the problem, we are given four symbols:
- △, □, ♦, and ♣, which represent **four different integers from 1 to 9**.
- Three equations involving these symbols:

$$
\begin{align*}
(1)\quad \triangle + \square &= \clubsuit \\
(2)\quad \triangle + \triangle &= \diamond + \diamond + \diamond + \diamond + \diamond \\
(3)\quad \triangle + \triangle &= \clubsuit + \diamond
\end{align*}
$$

---

### Step 1: Assign Variables

Let’s assign variables to the symbols for clarity:

- Let $ a = \triangle $
- Let $ b = \square $
- Let $ c = \diamond $
- Let $ d = \clubsuit $

Now rewrite the equations:

$$
\begin{align*}
(1)\quad a + b &= d \\
(2)\quad 2a &= 5c \\
(3)\quad 2a &= d + c
\end{align*}
$$

---

### Step 2: Use Equations (2) and (3) to Eliminate $ d $

From equations (2) and (3), both equal $ 2a $, so we can equate them:

$$
5c = d + c \Rightarrow 4c = d
$$

So, $ d = 4c $

---

### Step 3: Substitute into Equation (1)

From equation (1):
$$
a + b = d = 4c
$$

So,
$$
a + b = 4c \tag{4}
$$

From equation (2):
$$
2a = 5c \Rightarrow a = \frac{5c}{2}
$$

Since $ a $ must be an integer, $ 5c $ must be even. Since 5 is odd, $ c $ must be even.

---

### Step 4: Try Possible Values for $ c $

We know $ c \in \{1, 2, 3, ..., 9\} $, and $ c $ must be even. So possible values for $ c $ are:
$$
c = 2, 4, 6, 8
$$

Try each:

#### Case 1: $ c = 2 $

- $ a = \frac{5 \cdot 2}{2} = 5 $
- $ d = 4c = 8 $
- From equation (1): $ a + b = d \Rightarrow 5 + b = 8 \Rightarrow b = 3 $

So we have:
- $ a = 5 $
- $ b = 3 $
- $ c = 2 $
- $ d = 8 $

Check if all values are distinct and between 1 and 9:
✅ All values are distinct and in the range.

Now verify all equations:

- (1): $ a + b = 5 + 3 = 8 = d $ ✅
- (2): $ 2a = 10 $, and $ 5c = 10 $ ✅
- (3): $ 2a = 10 $, and $ d + c = 8 + 2 = 10 $ ✅

All equations are satisfied.

---

### Step 5: Check for Other Possible Values of $ c $

Try $ c = 4 $:
- $ a = \frac{5 \cdot 4}{2} = 10 $ → Invalid (not in 1–9)

Try $ c = 6 $:
- $ a = \frac{5 \cdot 6}{2} = 15 $ → Invalid

Try $ c = 8 $:
- $ a = \frac{5 \cdot 8}{2} = 20 $ → Invalid

So, only $ c = 2 $ gives valid values for all variables.

---

### Final Answer

From the above, the value of $ \square $ is:

$$
\boxed{3}
$$

Solution:
For simplicity, replace the triangle with the letter $a$, the square with the letter $b$, the diamond with the letter $c$, and the club with the letter $d$. The three given equations become \begin{align*}
a+b&=d\\
2a&=5c\\
2a&=c+d
\end{align*} We want to find the value of $b$. We can substitute the second equation into the third equation to eliminate $a$, to get $5c=c+d \Rightarrow 4c=d$. Since $a$, $b$, $c$, and $d$ are all integers from 1 to 9, we know that $d$ must be either 4 or 8 and $c$ correspondingly either 1 or 2. The first case, $c=1$ and $d=4$, does not work because plugging those two values into the third given equation gives $2a=5$, which is impossible if $a$ is an integer. Thus, $c=2$ and $d=8$. Plugging these values into the third given equation to solve for $a$, we have $2a=2+8\Rightarrow a=5$. Plugging $a=5$ and $d=8$ into the first equation to solve for $b$, we have $5+b=8 \Rightarrow b=3$. Thus, the value of the square is $\boxed{3}$.


Explanation:
Okay, so I need to find the quotient when x⁹ + 1 is divided by x - 1. Hmm, let me think. I remember that when you divide a polynomial by a linear term like x - a, you can use polynomial long division or maybe synthetic division. But I also recall something about the Remainder Theorem, which says that if you divide a polynomial f(x) by x - a, the remainder is f(a). But the question isn't asking for the remainder; it's asking for the quotient. So maybe I need to do the division?

Wait, but maybe there's a smarter way than doing long division? Let me think. If I can factor x⁹ + 1, maybe that would help? Let me recall that xⁿ + 1 can be factored if n is odd. Since 9 is odd, x⁹ + 1 can be factored as (x + 1)(x⁸ - x⁷ + x⁶ - x⁵ + x⁴ - x³ + x² - x + 1). Is that right? Let me check with a smaller exponent. For example, x³ + 1 = (x + 1)(x² - x + 1). Yeah, that works. So similarly, x⁹ + 1 would be (x + 1) times some polynomial. But the divisor here is x - 1, not x + 1. So maybe that's not directly helpful?

Wait, but maybe I can use polynomial division. Let me try that. Dividing x⁹ + 1 by x - 1. Let me recall how polynomial long division works. You divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient. Then multiply the entire divisor by that term, subtract from the dividend, and repeat.

So, let's set it up. Dividing x⁹ + 0x⁸ + 0x⁷ + 0x⁶ + 0x⁵ + 0x⁴ + 0x³ + 0x² + 0x + 1 by x - 1.

First term: x⁹ divided by x is x⁸. Multiply (x - 1) by x⁸: x⁹ - x⁸. Subtract that from the dividend:

(x⁹ + 0x⁸ + ...) - (x⁹ - x⁸) = 0x⁹ + x⁸ + 0x⁷ + ... So the new dividend is x⁸ + 0x⁷ + 0x⁶ + 0x⁵ + 0x⁴ + 0x³ + 0x² + 0x + 1.

Next term: x⁸ divided by x is x⁷. Multiply (x - 1) by x⁷: x⁸ - x⁷. Subtract that from the current dividend:

(x⁸ + 0x⁷ + ...) - (x⁸ - x⁷) = 0x⁸ + x⁷ + 0x⁶ + ... So now the dividend is x⁷ + 0x⁶ + 0x⁵ + 0x⁴ + 0x³ + 0x² + 0x + 1.

Continuing this pattern, the next term would be x⁷ divided by x is x⁶. Multiply (x - 1) by x⁶: x⁷ - x⁶. Subtract: (x⁷ + 0x⁶ + ...) - (x⁷ - x⁶) = 0x⁷ + x⁶ + ... So the dividend becomes x⁶ + 0x⁵ + 0x⁴ + 0x³ + 0x² + 0x + 1.

This seems like a pattern. Each time, the next term is x^(n) where n decreases by 1 each time, and the remainder alternates signs? Wait, but the dividend is always adding a term. Let me see:

After the first subtraction, we had x⁸ + ... Then after subtracting, we had x⁷ + ... Then x⁶ + ... So each time, the degree reduces by 1. So continuing this process, the quotient terms would be x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1? Wait, but let me check.

Wait, if we keep doing this, after subtracting each time, the next term is positive. Let me think. Let me try to do a few more steps.

After the third step, we had x⁶ + ... Then next term is x⁶ / x = x⁵. Multiply (x - 1) by x⁵: x⁶ - x⁵. Subtract: (x⁶ + 0x⁵ + ...) - (x⁶ - x⁵) = 0x⁶ + x⁵ + ... So the dividend becomes x⁵ + ... So the quotient so far is x⁸ + x⁷ + x⁶ + x⁵.

Continuing this, the next term would be x⁵ / x = x⁴. Multiply (x - 1) by x⁴: x⁵ - x⁴. Subtract: (x⁵ + 0x⁴ + ...) - (x⁵ - x⁴) = 0x⁵ + x⁴ + ... So the dividend is x⁴ + ... Quotient now is x⁸ + x⁷ + x⁶ + x⁵ + x⁴.

Continuing this pattern, each time we add another term to the quotient, and the dividend becomes the next lower degree term. So continuing until we get down to the constant term.

So after all these steps, the quotient would be x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1, and then the remainder would be... Let me check. When we get down to the last step, the dividend would be 1 (the constant term). Then we divide 1 by x - 1, but since the degree of the dividend is 0 and the divisor is degree 1, we can't divide further. So the remainder is 1. Wait, but according to the Remainder Theorem, if we plug x = 1 into the original polynomial, we get 1⁹ + 1 = 2. So the remainder should be 2. But according to my division steps, the remainder is 1? That's a contradiction. So I must have made a mistake in my division steps.

Wait, let me check again. Let me try to do the division more carefully. Let me write down the steps again.

Dividend: x⁹ + 1. Let me write all the terms:

x⁹ + 0x⁸ + 0x⁷ + 0x⁶ + 0x⁵ + 0x⁴ + 0x³ + 0x² + 0x + 1.

Divisor: x - 1.

First term: x⁹ / x = x⁸. Multiply divisor by x⁸: x⁸*(x - 1) = x⁹ - x⁸. Subtract this from the dividend:

(x⁹ + 0x⁸ + ...) - (x⁹ - x⁸) = 0x⁹ + x⁸ + 0x⁷ + 0x⁶ + 0x⁵ + 0x⁴ + 0x³ + 0x² + 0x + 1.

So new dividend: x⁸ + 0x⁷ + 0x⁶ + 0x⁵ + 0x⁴ + 0x³ + 0x² + 0x + 1.

Second term: x⁸ / x = x⁷. Multiply divisor by x⁷: x⁷*(x - 1) = x⁸ - x⁷. Subtract:

(x⁸ + 0x⁷ + ...) - (x⁸ - x⁷) = 0x⁸ + x⁷ + 0x⁶ + 0x⁵ + 0x⁴ + 0x³ + 0x² + 0x + 1.

New dividend: x⁷ + 0x⁶ + 0x⁵ + 0x⁴ + 0x³ + 0x² + 0x + 1.

Third term: x⁷ / x = x⁶. Multiply divisor by x⁶: x⁶*(x - 1) = x⁷ - x⁶. Subtract:

(x⁷ + 0x⁶ + ...) - (x⁷ - x⁶) = 0x⁷ + x⁶ + 0x⁵ + 0x⁴ + 0x³ + 0x² + 0x + 1.

New dividend: x⁶ + 0x⁵ + 0x⁴ + 0x³ + 0x² + 0x + 1.

Fourth term: x⁶ / x = x⁵. Multiply divisor by x⁵: x⁵*(x - 1) = x⁶ - x⁵. Subtract:

(x⁶ + 0x⁵ + ...) - (x⁶ - x⁵) = 0x⁶ + x⁵ + 0x⁴ + 0x³ + 0x² + 0x + 1.

New dividend: x⁵ + 0x⁴ + 0x³ + 0x² + 0x + 1.

Fifth term: x⁵ / x = x⁴. Multiply divisor by x⁴: x⁴*(x - 1) = x⁵ - x⁴. Subtract:

(x⁵ + 0x⁴ + ...) - (x⁵ - x⁴) = 0x⁵ + x⁴ + 0x³ + 0x² + 0x + 1.

New dividend: x⁴ + 0x³ + 0x² + 0x + 1.

Sixth term: x⁴ / x = x³. Multiply divisor by x³: x³*(x - 1) = x⁴ - x³. Subtract:

(x⁴ + 0x³ + ...) - (x⁴ - x³) = 0x⁴ + x³ + 0x² + 0x + 1.

New dividend: x³ + 0x² + 0x + 1.

Seventh term: x³ / x = x². Multiply divisor by x²: x²*(x - 1) = x³ - x². Subtract:

(x³ + 0x² + ...) - (x³ - x²) = 0x³ + x² + 0x + 1.

New dividend: x² + 0x + 1.

Eighth term: x² / x = x. Multiply divisor by x: x*(x - 1) = x² - x. Subtract:

(x² + 0x + 1) - (x² - x) = 0x² + x + 1.

New dividend: x + 1.

Ninth term: x / x = 1. Multiply divisor by 1: 1*(x - 1) = x - 1. Subtract:

(x + 1) - (x - 1) = 0x + 2.

So the remainder is 2, which matches the Remainder Theorem. So the quotient is the sum of all the terms we found during division: x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1. And the remainder is 2. Therefore, the quotient is that polynomial.

Wait, but let me check if that's correct. Let me multiply (x - 1) by the quotient and see if I get x⁹ + 1.

Let me compute (x - 1)(x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1). Let me distribute:

First, x*(x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1) = x⁹ + x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x.

Then subtract 1*(x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1) = -x⁸ - x⁷ - x⁶ - x⁵ - x⁴ - x³ - x² - x - 1.

Adding these two results together:

x⁹ + x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x - x⁸ - x⁷ - x⁶ - x⁵ - x⁴ - x³ - x² - x - 1.

Simplify term by term:

x⁹ + (x⁸ - x⁸) + (x⁷ - x⁷) + (x⁶ - x⁶) + (x⁵ - x⁵) + (x⁴ - x⁴) + (x³ - x³) + (x² - x²) + (x - x) - 1.

All the middle terms cancel out, leaving x⁹ - 1. Wait, but the original polynomial was x⁹ + 1. So according to this, (x - 1)*quotient = x⁹ - 1. But we have x⁹ + 1. So there's a discrepancy here. That means I must have made a mistake in my division steps. Wait, but earlier when I did the division, the remainder was 2. So according to division algorithm, dividend = divisor * quotient + remainder. So x⁹ + 1 = (x - 1)*quotient + 2. Therefore, (x - 1)*quotient = x⁹ + 1 - 2 = x⁹ - 1. Which matches the multiplication above. So that's correct. Therefore, the quotient is indeed x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1, and the remainder is 2. Therefore, the answer to the question is that quotient.

But let me check again with another method. Maybe using the formula for geometric series? Because x⁹ + 1 can be thought of as x⁹ - (-1). So if I consider the sum of a geometric series. Let me recall that xⁿ - 1 = (x - 1)(xⁿ⁻¹ + xⁿ⁻² + ... + x + 1). So if I have x⁹ - 1, that factors as (x - 1)(x⁸ + x⁷ + ... + 1). But our original polynomial is x⁹ + 1, which is x⁹ - (-1). So if I want to write x⁹ + 1 as (x - 1)*quotient + remainder, but how does that relate?

Alternatively, maybe I can write x⁹ + 1 = x⁹ - 1 + 2. Then, since x⁹ - 1 = (x - 1)(x⁸ + x⁷ + ... + 1), so x⁹ + 1 = (x - 1)(x⁸ + x⁷ + ... + 1) + 2. Therefore, the quotient is x⁸ + x⁷ + ... + 1 and the remainder is 2. Which matches what I found earlier. Therefore, the quotient is indeed the sum from k=0 to 8 of x^k, which is x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1.

Alternatively, maybe there's a formula for dividing xⁿ + 1 by x - 1. Let me think. If n is odd, then xⁿ + 1 can be factored as (x + 1)(xⁿ⁻¹ - xⁿ⁻² + xⁿ⁻³ - ... + 1). But that's different from dividing by x - 1. However, if I use the formula for the sum of a geometric series. Let me recall that the sum of a geometric series with ratio x is (xⁿ - 1)/(x - 1). But if I have xⁿ + 1, that's similar to (xⁿ - (-1)) / (x - 1). So if I write xⁿ + 1 = (xⁿ - (-1)) / (x - 1) * (x - 1) + remainder. Wait, but that's not helpful. Wait, but if I consider that (xⁿ - 1)/(x - 1) = xⁿ⁻¹ + xⁿ⁻² + ... + 1. So if I have xⁿ + 1, which is xⁿ - (-1). So if I want to write xⁿ + 1 divided by x - 1, then it's equal to [(xⁿ - 1) + 2]/(x - 1) = (xⁿ - 1)/(x - 1) + 2/(x - 1). Therefore, the quotient is (xⁿ - 1)/(x - 1) and the remainder is 2. Which again gives the same result. Therefore, for x⁹ + 1 divided by x - 1, the quotient is (x⁹ - 1)/(x - 1) = x⁸ + x⁷ + ... + 1, and the remainder is 2. Therefore, the answer is that quotient.

So, the quotient is x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1. Let me check if that's the case. Let me plug in x = 2. Let me compute (2⁹ + 1) divided by (2 - 1) = 513 / 1 = 513. On the other hand, the quotient evaluated at x = 2 is 2⁸ + 2⁷ + 2⁶ + 2⁵ + 2⁴ + 2³ + 2² + 2 + 1. Let me compute that:

2⁸ = 256

2⁷ = 128

2⁶ = 64

2⁵ = 32

2⁴ = 16

2³ = 8

2² = 4

2 = 2

1 = 1

Adding them up: 256 + 128 = 384; 384 + 64 = 448; 448 + 32 = 480; 480 + 16 = 496; 496 + 8 = 504; 504 + 4 = 508; 508 + 2 = 510; 510 + 1 = 511. Then, according to the division algorithm, dividend = divisor * quotient + remainder. So 513 = (2 - 1)*511 + remainder. Which is 513 = 1*511 + remainder => remainder = 2. Which matches. Therefore, the quotient is indeed 511 when x = 2, which is the sum of the geometric series. Therefore, the quotient is correct.

Therefore, the answer is x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1. So I can write that as a polynomial. Alternatively, maybe there's a more compact way to write it? Like using summation notation or something? But the question says "find the quotient," and unless it specifies a particular form, the expanded form is probably acceptable.

Alternatively, since the quotient is a geometric series with 9 terms (from x⁰ to x⁸), which is (x⁹ - 1)/(x - 1). But since we are dividing x⁹ + 1 by x - 1, the quotient is (x⁹ - 1)/(x - 1) + 2/(x - 1). Wait, but that's not helpful. However, since we already know that (x⁹ - 1)/(x - 1) is the quotient when dividing x⁹ - 1 by x - 1, which is the same as the quotient here except for the remainder. But since we have x⁹ + 1 = (x - 1)*quotient + 2, then quotient = (x⁹ + 1 - 2)/(x - 1) = (x⁹ - 1)/(x - 1). Wait, that's interesting. So quotient is (x⁹ - 1)/(x - 1). Which is the same as the sum from k=0 to 8 of x^k. Therefore, the quotient is that sum. Therefore, the answer is x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1.

Therefore, I think that's the correct quotient. Let me check once more with another value. Let me take x = 0. Dividing 0⁹ + 1 = 1 by 0 - 1 = -1. So the quotient should be 1 / (-1) = -1, but with remainder? Wait, according to the division algorithm, dividend = divisor * quotient + remainder. So 1 = (-1)*quotient + remainder. But if x = 0, then the original division is 1 divided by -1, which is -1 with remainder 0. Wait, but according to our quotient polynomial, when x = 0, the quotient is 0⁸ + 0⁷ + ... + 1 = 1. Then, according to the division algorithm, 1 = (-1)*1 + remainder => remainder = 1 - (-1)*1 = 1 + 1 = 2? Wait, but when x = 0, the original polynomial is 1, and divisor is -1, so 1 divided by -1 is -1 with remainder 0. But according to our previous calculation, the remainder is 2. Wait, that seems conflicting. Wait, no, because when we do polynomial division, the remainder is a constant, but when we plug in x = 0, the remainder is 2, but according to the division algorithm, the remainder is 2. However, when we evaluate the equation x⁹ + 1 = (x - 1)*quotient + 2 at x = 0, we get 0 + 1 = (-1)*quotient(0) + 2. Therefore, 1 = (-1)*1 + 2 => 1 = -1 + 2 => 1 = 1. Which is correct. So the remainder is 2, which is consistent. Therefore, even though when x = 0, the division of 1 by -1 would give quotient -1 and remainder 0, but in polynomial division, the remainder is 2, which is a constant, not dependent on x. Therefore, the quotient polynomial is correct.

Therefore, after all these checks, I think the quotient is indeed x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1.

**Final Answer**
The quotient is \boxed{x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1}.

To find the quotient when $ x^9 + 1 $ is divided by $ x - 1 $, we can use the **polynomial division algorithm** or a more elegant approach using the **geometric series**.

---

### Step 1: Understanding the Division

We are dividing the polynomial $ x^9 + 1 $ by $ x - 1 $. The **Remainder Theorem** tells us that the remainder of this division is:

$$
f(1) = 1^9 + 1 = 2
$$

So, the division can be written as:

$$
x^9 + 1 = (x - 1) \cdot Q(x) + 2
$$

Where $ Q(x) $ is the **quotient** we are trying to find.

---

### Step 2: Using the Geometric Series

We can rewrite the expression $ x^9 + 1 $ as:

$$
x^9 + 1 = x^9 - (-1)
$$

This resembles the form of a geometric series:

$$
x^9 - 1 = (x - 1)(x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)
$$

So, we can write:

$$
x^9 + 1 = (x - 1)(x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) + 2
$$

This confirms that the **quotient** is:

$$
x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
$$

---

### Step 3: Final Answer

Thus, the quotient when $ x^9 + 1 $ is divided by $ x - 1 $ is:

$$
\boxed{x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1}
$$

[Assistant]
Explanation:
Okay, so I need to find the quotient when x⁹ + 1 is divided by x - 1. Hmm, let me think. I remember that when you divide a polynomial by a linear term like x - a, you can use polynomial long division or maybe synthetic division. But I also recall something about the Remainder Theorem, which says that if you divide a polynomial f(x) by x - a, the remainder is f(a). But the question isn't asking for the remainder; it's asking for the quotient. So maybe I need to do the division?

Wait, but maybe there's a smarter way than doing long division? Let me think. If I can factor x⁹ + 1, maybe that would help? Let me recall that xⁿ + 1 can be factored if n is odd. Since 9 is odd, x⁹ + 1 can be factored as (x + 1)(x⁸ - x⁷ + x⁶ - x⁵ + x⁴ - x³ + x² - x + 1). Is that right? Let me check with a smaller exponent. For example, x³ + 1 = (x + 1)(x² - x + 1). Yeah, that works. So similarly, x⁹ + 1 would be (x + 1) times some polynomial. But the divisor here is x - 1, not x + 1. So maybe that's not directly helpful?

Wait, but maybe I can use polynomial division. Let me try that. Dividing x⁹ + 1 by x - 1. Let me recall how polynomial long division works. You divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient. Then multiply the entire divisor by that term, subtract from the dividend, and repeat.

So, let's set it up. Dividing x⁹ + 0x⁸ + 0x⁷ + 0x⁶ + 0x⁵ + 0x⁴ + 0x³ + 0x² + 0x + 1 by x - 1.

First term: x⁹ divided by x is x⁸. Multiply (x - 1) by x⁸: x⁹ - x⁸. Subtract that from the dividend:

(x⁹ + 0x⁸ + ...) - (x⁹ - x⁸) = 0x⁹ + x⁸ + 0x⁷ + ... So the new dividend is x⁸ + 0x⁷ + 0x⁶ + 0x⁵ + 0x⁴ + 0x³ + 0x² + 0x + 1.

Next term: x⁸ divided by x is x⁷. Multiply (x - 1) by x⁷: x⁸ - x⁷. Subtract that from the current dividend:

(x⁸ + 0x⁷ + ...) - (x⁸ - x⁷) = 0x⁸ + x⁷ + 0x⁶ + ... So now the dividend is x⁷ + 0x⁶ + 0x⁵ + 0x⁴ + 0x³ + 0x² + 0x + 1.

Continuing this pattern, the next term would be x⁷ divided by x is x⁶. Multiply (x - 1) by x⁶: x⁷ - x⁶. Subtract: (x⁷ + 0x⁶ + ...) - (x⁷ - x⁶) = 0x⁷ + x⁶ + ... So the dividend becomes x⁶ + 0x⁵ + 0x⁴ + 0x³ + 0x² + 0x + 1.

This seems like a pattern. Each time, the next term is x^(n) where n decreases by 1 each time, and the remainder alternates signs? Wait, but the dividend is always adding a term. Let me see:

After the first subtraction, we had x⁸ + ... Then after subtracting, we had x⁷ + ... Then x⁶ + ... So each time, the degree reduces by 1. So continuing this process, the quotient terms would be x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1? Wait, but let me check.

Wait, if we keep doing this, after subtracting each time, the next term is positive. Let me think. Let me try to do a few more steps.

After the third step, we had x⁶ + ... Then next term is x⁶ / x = x⁵. Multiply (x - 1) by x⁵: x⁶ - x⁵. Subtract: (x⁶ + 0x⁵ + ...) - (x⁶ - x⁵) = 0x⁶ + x⁵ + ... So the dividend becomes x⁵ + ... So the quotient so far is x⁸ + x⁷ + x⁶ + x⁵.

Continuing this, the next term would be x⁵ / x = x⁴. Multiply (x - 1) by x⁴: x⁵ - x⁴. Subtract: (x⁵ + 0x⁴ + ...) - (x⁵ - x⁴) = 0x⁵ + x⁴ + ... So the dividend is x⁴ + ... Quotient now is x⁸ + x⁷ + x⁶ + x⁵ + x⁴.

Continuing this pattern, each time we add another term to the quotient, and the dividend becomes the next lower degree term. So continuing until we get down to the constant term.

So after all these steps, the quotient would be x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1, and then the remainder would be... Let me check. When we get down to the last step, the dividend would be 1 (the constant term). Then we divide 1 by x - 1, but since the degree of the dividend is 0 and the divisor is degree 1, we can't divide further. So the remainder is 1. Wait, but according to the Remainder Theorem, if we plug x = 1 into the original polynomial, we get 1⁹ + 1 = 2. So the remainder should be 2. But according to my division steps, the remainder is 1? That's a contradiction. So I must have made a mistake in my division steps.

Wait, let me check again. Let me try to do the division more carefully. Let me write down the steps again.

Dividend: x⁹ + 1. Let me write all the terms:

x⁹ + 0x⁸ + 0x⁷ + 0x⁶ + 0x⁵ + 0x⁴ + 0x³ + 0x² + 0x + 1.

Divisor: x - 1.

First term: x⁹ / x = x⁸. Multiply divisor by x⁸: x⁸*(x - 1) = x⁹ - x⁸. Subtract this from the dividend:

(x⁹ + 0x⁸ + ...) - (x⁹ - x⁸) = 0x⁹ + x⁸ + 0x⁷ + 0x⁶ + 0x⁵ + 0x⁴ + 0x³ + 0x² + 0x + 1.

So new dividend: x⁸ + 0x⁷ + 0x⁶ + 0x⁵ + 0x⁴ + 0x³ + 0x² + 0x + 1.

Second term: x⁸ / x = x⁷. Multiply divisor by x⁷: x⁷*(x - 1) = x⁸ - x⁷. Subtract:

(x⁸ + 0x⁷ + ...) - (x⁸ - x⁷) = 0x⁸ + x⁷ + 0x⁶ + 0x⁵ + 0x⁴ + 0x³ + 0x² + 0x + 1.

New dividend: x⁷ + 0x⁶ + 0x⁵ + 0x⁴ + 0x³ + 0x² + 0x + 1.

Third term: x⁷ / x = x⁶. Multiply divisor by x⁶: x⁶*(x - 1) = x⁷ - x⁶. Subtract:

(x⁷ + 0x⁶ + ...) - (x⁷ - x⁶) = 0x⁷ + x⁶ + 0x⁵ + 0x⁴ + 0x³ + 0x² + 0x + 1.

New dividend: x⁶ + 0x⁵ + 0x⁴ + 0x³ + 0x² + 0x + 1.

Fourth term: x⁶ / x = x⁵. Multiply divisor by x⁵: x⁵*(x - 1) = x⁶ - x⁵. Subtract:

(x⁶ + 0x⁵ + ...) - (x⁶ - x⁵) = 0x⁶ + x⁵ + 0x⁴ + 0x³ + 0x² + 0x + 1.

New dividend: x⁵ + 0x⁴ + 0x³ + 0x² + 0x + 1.

Fifth term: x⁵ / x = x⁴. Multiply divisor by x⁴: x⁴*(x - 1) = x⁵ - x⁴. Subtract:

(x⁵ + 0x⁴ + ...) - (x⁵ - x⁴) = 0x⁵ + x⁴ + 0x³ + 0x² + 0x + 1.

New dividend: x⁴ + 0x³ + 0x² + 0x + 1.

Sixth term: x⁴ / x = x³. Multiply divisor by x³: x³*(x - 1) = x⁴ - x³. Subtract:

(x⁴ + 0x³ + ...) - (x⁴ - x³) = 0x⁴ + x³ + 0x² + 0x + 1.

New dividend: x³ + 0x² + 0x + 1.

Seventh term: x³ / x = x². Multiply divisor by x²: x²*(x - 1) = x³ - x². Subtract:

(x³ + 0x² + ...) - (x³ - x²) = 0x³ + x² + 0x + 1.

New dividend: x² + 0x + 1.

Eighth term: x² / x = x. Multiply divisor by x: x*(x - 1) = x² - x. Subtract:

(x² + 0x + 1) - (x² - x) = 0x² + x + 1.

New dividend: x + 1.

Ninth term: x / x = 1. Multiply divisor by 1: 1*(x - 1) = x - 1. Subtract:

(x + 1) - (x - 1) = 0x + 2.

So the remainder is 2, which matches the Remainder Theorem. So the quotient is the sum of all the terms we found during division: x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1. And the remainder is 2. Therefore, the quotient is that polynomial.

Wait, but let me check if that's correct. Let me multiply (x - 1) by the quotient and see if I get x⁹ + 1.

Let me compute (x - 1)(x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1). Let me distribute:

First, x*(x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1) = x⁹ + x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x.

Then subtract 1*(x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1) = -x⁸ - x⁷ - x⁶ - x⁵ - x⁴ - x³ - x² - x - 1.

Adding these two results together:

x⁹ + x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x - x⁸ - x⁷ - x⁶ - x⁵ - x⁴ - x³ - x² - x - 1.

Simplify term by term:

x⁹ + (x⁸ - x⁸) + (x⁷ - x⁷) + (x⁶ - x⁶) + (x⁵ - x⁵) + (x⁴ - x⁴) + (x³ - x³) + (x² - x²) + (x - x) - 1.

All the middle terms cancel out, leaving x⁹ - 1. Wait, but the original polynomial was x⁹ + 1. So according to this, (x - 1)*quotient = x⁹ - 1. But we have x⁹ + 1. So there's a discrepancy here. That means I must have made a mistake in my division steps. Wait, but earlier when I did the division, the remainder was 2. So according to division algorithm, dividend = divisor * quotient + remainder. So x⁹ + 1 = (x - 1)*quotient + 2. Therefore, (x - 1)*quotient = x⁹ + 1 - 2 = x⁹ - 1. Which matches the multiplication above. So that's correct. Therefore, the quotient is indeed x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1, and the remainder is 2. Therefore, the answer to the question is that quotient.

But let me check again with another method. Maybe using the formula for geometric series? Because x⁹ + 1 can be thought of as x⁹ - (-1). So if I consider the sum of a geometric series. Let me recall that xⁿ - 1 = (x - 1)(xⁿ⁻¹ + xⁿ⁻² + ... + x + 1). So if I have x⁹ - 1, that factors as (x - 1)(x⁸ + x⁷ + ... + 1). But our original polynomial is x⁹ + 1, which is x⁹ - (-1). So if I want to write x⁹ + 1 as (x - 1)*quotient + remainder, but how does that relate?

Alternatively, maybe I can write x⁹ + 1 = x⁹ - 1 + 2. Then, since x⁹ - 1 = (x - 1)(x⁸ + x⁷ + ... + 1), so x⁹ + 1 = (x - 1)(x⁸ + x⁷ + ... + 1) + 2. Therefore, the quotient is x⁸ + x⁷ + ... + 1 and the remainder is 2. Which matches what I found earlier. Therefore, the quotient is indeed the sum from k=0 to 8 of x^k, which is x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1.

Alternatively, maybe there's a formula for dividing xⁿ + 1 by x - 1. Let me think. If n is odd, then xⁿ + 1 can be factored as (x + 1)(xⁿ⁻¹ - xⁿ⁻² + xⁿ⁻³ - ... + 1). But that's different from dividing by x - 1. However, if I use the formula for the sum of a geometric series. Let me recall that the sum of a geometric series with ratio x is (xⁿ - 1)/(x - 1). But if I have xⁿ + 1, that's similar to (xⁿ - (-1)) / (x - 1). So if I write xⁿ + 1 = (xⁿ - (-1)) / (x - 1) * (x - 1) + remainder. Wait, but that's not helpful. Wait, but if I consider that (xⁿ - 1)/(x - 1) = xⁿ⁻¹ + xⁿ⁻² + ... + 1. So if I have xⁿ + 1, which is xⁿ - (-1). So if I want to write xⁿ + 1 divided by x - 1, then it's equal to [(xⁿ - 1) + 2]/(x - 1) = (xⁿ - 1)/(x - 1) + 2/(x - 1). Therefore, the quotient is (xⁿ - 1)/(x - 1) and the remainder is 2. Which again gives the same result. Therefore, for x⁹ + 1 divided by x - 1, the quotient is (x⁹ - 1)/(x - 1) = x⁸ + x⁷ + ... + 1, and the remainder is 2. Therefore, the answer is that quotient.

So, the quotient is x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1. Let me check if that's the case. Let me plug in x = 2. Let me compute (2⁹ + 1) divided by (2 - 1) = 513 / 1 = 513. On the other hand, the quotient evaluated at x = 2 is 2⁸ + 2⁷ + 2⁶ + 2⁵ + 2⁴ + 2³ + 2² + 2 + 1. Let me compute that:

2⁸ = 256

2⁷ = 128

2⁶ = 64

2⁵ = 32

2⁴ = 16

2³ = 8

2² = 4

2 = 2

1 = 1

Adding them up: 256 + 128 = 384; 384 + 64 = 448; 448 + 32 = 480; 480 + 16 = 496; 496 + 8 = 504; 504 + 4 = 508; 508 + 2 = 510; 510 + 1 = 511. Then, according to the division algorithm, dividend = divisor * quotient + remainder. So 513 = (2 - 1)*511 + remainder. Which is 513 = 1*511 + remainder => remainder = 2. Which matches. Therefore, the quotient is indeed 511 when x = 2, which is the sum of the geometric series. Therefore, the quotient is correct.

Therefore, the answer is x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1. So I can write that as a polynomial. Alternatively, maybe there's a more compact way to write it? Like using summation notation or something? But the question says "find the quotient," and unless it specifies a particular form, the expanded form is probably acceptable.

Alternatively, since the quotient is a geometric series with 9 terms (from x⁰ to x⁸), which is (x⁹ - 1)/(x - 1). But since we are dividing x⁹ + 1 by x - 1, the quotient is (x⁹ - 1)/(x - 1) + 2/(x - 1). Wait, but that's not helpful. However, since we already know that (x⁹ - 1)/(x - 1) is the quotient when dividing x⁹ - 1 by x - 1, which is the same as the quotient here except for the remainder. But since we have x⁹ + 1 = (x - 1)*quotient + 2, then quotient = (x⁹ + 1 - 2)/(x - 1) = (x⁹ - 1)/(x - 1). Wait, that's interesting. So quotient is (x⁹ - 1)/(x - 1). Which is the same as the sum from k=0 to 8 of x^k. Therefore, the quotient is that sum. Therefore, the answer is x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1.

Therefore, I think that's the correct quotient. Let me check once more with another value. Let me take x = 0. Dividing 0⁹ + 1 = 1 by 0 - 1 = -1. So the quotient should be 1 / (-1) = -1, but with remainder? Wait, according to the division algorithm, dividend = divisor * quotient + remainder. So 1 = (-1)*quotient + remainder. But if x = 0, then the original division is 1 divided by -1, which is -1 with remainder 0. Wait, but according to our quotient polynomial, when x = 0, the quotient is 0⁸ + 0⁷ + ... + 1 = 1. Then, according to the division algorithm, 1 = (-1)*1 + remainder => remainder = 1 - (-1)*1 = 1 + 1 = 2? Wait, but when x = 0, the original polynomial is 1, and divisor is -1, so 1 divided by -1 is -1 with remainder 0. But according to our previous calculation, the remainder is 2. Wait, that seems conflicting. Wait, no, because when we do polynomial division, the remainder is a constant, but when we plug in x = 0, the remainder is 2, but according to the division algorithm, the remainder is 2. However, when we evaluate the equation x⁹ + 1 = (x - 1)*quotient + 2 at x = 0, we get 0 + 1 = (-1)*quotient(0) + 2. Therefore, 1 = (-1)*1 + 2 => 1 = -1 + 2 => 1 = 1. Which is correct. So the remainder is 2, which is consistent. Therefore, even though when x = 0, the division of 1 by -1 would give quotient -1 and remainder 0, but in polynomial division, the remainder is 2, which is a constant, not dependent on x. Therefore, the quotient polynomial is correct.

Therefore, after all these checks, I think the quotient is indeed x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1.

**Final Answer**
The quotient is \boxed{x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1}.

To find the quotient when $ x^9 + 1 $ is divided by $ x - 1 $, we can use the **polynomial division algorithm** or a more elegant approach using the **geometric series**.

---

### Step 1: Understanding the Division

We are dividing the polynomial $ x^9 + 1 $ by $ x - 1 $. The **Remainder Theorem** tells us that the remainder of this division is:

$$
f(1) = 1^9 + 1 = 2
$$

So, the division can be written as:

$$
x^9 + 1 = (x - 1) \cdot Q(x) + 2
$$

Where $ Q(x) $ is the **quotient** we are trying to find.

---

### Step 2: Using the Geometric Series

We can rewrite the expression $ x^9 + 1 $ as:

$$
x^9 + 1 = x^9 - (-1)
$$

This resembles the form of a geometric series:

$$
x^9 - 1 = (x - 1)(x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)
$$

So, we can write:

$$
x^9 + 1 = (x - 1)(x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) + 2
$$

This confirms that the **quotient** is:

$$
x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
$$

---

### Step 3: Final Answer

Thus, the quotient when $ x^9 + 1 $ is divided by $ x - 1 $ is:

$$
\boxed{x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1}
$$

Solution:
We can perform long division.  We can also write
\begin{align*}
\frac{x^9 + 1}{x - 1} &= \frac{(x^9 - 1) + 2}{x - 1} \\
&= \frac{x^9 - 1}{x - 1} + \frac{2}{x - 1} \\
&= x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 + \frac{2}{x - 1}.
\end{align*}Thus, the quotient is $\boxed{x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1}.$


Explanation:
Okay, so I need to simplify this expression: (u + 4)(u - 1) minus (u - 3)(u + 6). Let me think about how to approach this. I remember that to simplify expressions like this, I should probably expand both products first and then combine like terms. Let me start with the first part, (u + 4)(u - 1).

Alright, expanding that. I think I need to use the distributive property, also known as the FOIL method for binomials. Let me recall: FOIL stands for First, Outer, Inner, Last. So, for (u + 4)(u - 1):

First terms: u * u = u²

Outer terms: u * (-1) = -u

Inner terms: 4 * u = 4u

Last terms: 4 * (-1) = -4

Now, adding all those together: u² - u + 4u - 4. Let me combine like terms here. The -u and +4u can be combined. That would be (-1u + 4u) which is 3u. So the first part simplifies to u² + 3u - 4. Let me check that again.

Wait, u² - u + 4u - 4. Yes, combining -u + 4u is 3u. So that's correct. So the first part is u² + 3u - 4. Okay, got that.

Now, moving on to the second part: (u - 3)(u + 6). Again, I need to expand this. Let me apply the same FOIL method.

First terms: u * u = u²

Outer terms: u * 6 = 6u

Inner terms: -3 * u = -3u

Last terms: -3 * 6 = -18

Adding those together: u² + 6u - 3u - 18. Combining like terms here: 6u - 3u is 3u. So this simplifies to u² + 3u - 18. Wait, that's interesting. Both the first and second parts have u² + 3u terms? Let me check again.

First part: (u + 4)(u - 1) = u² - u + 4u - 4 = u² + 3u - 4. Correct.

Second part: (u - 3)(u + 6) = u² + 6u - 3u - 18 = u² + 3u - 18. Yes, that's right. So both expanded forms have u² + 3u, but different constants.

Now, the original expression is the first part minus the second part. So that would be:

(u² + 3u - 4) - (u² + 3u - 18).

Wait, I need to be careful with the subtraction. The entire second part is being subtracted. So I need to distribute the negative sign to each term in the second polynomial. Let me write that out:

= u² + 3u - 4 - u² - 3u + 18.

Wait, let me check that again. If I have (A) - (B), where A is u² + 3u - 4 and B is u² + 3u - 18, then it's A - B = (u² + 3u - 4) - u² - 3u + 18. Wait, no. Wait, subtracting each term in B. So:

Original expression: (u² + 3u - 4) - (u² + 3u - 18)

Which is equal to u² + 3u - 4 - u² - 3u + 18. Because subtracting u² is -u², subtracting 3u is -3u, and subtracting (-18) is +18.

Now, let me combine like terms here. Let's see:

u² - u² = 0.

3u - 3u = 0.

Then, -4 + 18 = 14.

So all the terms cancel out except for 14? That seems surprising. Let me check my steps again to make sure I didn't make a mistake.

First expansion: (u + 4)(u - 1) = u² + 3u - 4. Correct.

Second expansion: (u - 3)(u + 6) = u² + 3u - 18. Correct.

Subtracting the second from the first: (u² + 3u - 4) - (u² + 3u - 18).

Distribute the negative sign: u² + 3u - 4 - u² - 3u + 18.

Combine like terms:

u² - u² = 0.

3u - 3u = 0.

-4 + 18 = 14.

So the entire expression simplifies to 14? That seems too simple. Let me try plugging in a value for u to check. Let's pick u = 0.

Original expression: (0 + 4)(0 - 1) - (0 - 3)(0 + 6) = (4)(-1) - (-3)(6) = -4 - (-18) = -4 + 18 = 14.

Okay, that works. Let me try another value, say u = 1.

Original expression: (1 + 4)(1 - 1) - (1 - 3)(1 + 6) = (5)(0) - (-2)(7) = 0 - (-14) = 14.

Hmm, also 14. Let me try u = 2.

Original expression: (2 + 4)(2 - 1) - (2 - 3)(2 + 6) = (6)(1) - (-1)(8) = 6 - (-8) = 6 + 8 = 14.

Same result. So it seems like regardless of the value of u, the expression simplifies to 14. Therefore, my simplification is correct.

But just to make sure, let me try expanding again step by step.

First part: (u + 4)(u - 1). Multiply term by term:

u*(u) + u*(-1) + 4*(u) + 4*(-1) = u² - u + 4u - 4 = u² + 3u - 4. Correct.

Second part: (u - 3)(u + 6). Multiply term by term:

u*(u) + u*(6) + (-3)*(u) + (-3)*(6) = u² + 6u - 3u - 18 = u² + 3u - 18. Correct.

Subtracting the second from the first:

(u² + 3u - 4) - (u² + 3u - 18) = u² + 3u - 4 - u² - 3u + 18.

Combine like terms:

u² - u² = 0.

3u - 3u = 0.

-4 + 18 = 14.

Yes, that's correct. So the simplified form is 14.

Therefore, the answer is 14.

But just to make sure there isn't a trick or something I missed. The problem says "simplify," and 14 is a constant, so that's as simplified as it gets.

Alternatively, maybe there's another way to approach this problem? Let me think. Maybe by combining the two expressions before expanding? Let me try that.

Original expression: (u + 4)(u - 1) - (u - 3)(u + 6).

Alternatively, maybe factor something out? But since they are both quadratic terms, perhaps not. Alternatively, maybe notice that both have u² + 3u terms? Wait, when I expanded them, both had u² + 3u, so when subtracting, they cancel.

So, if I think of it as [(u² + 3u) - 4] - [(u² + 3u) - 18] = (u² + 3u) - 4 - u² - 3u + 18 = (-4 + 18) = 14.

Yes, that's another way to see it. The u² and 3u terms cancel out, leaving just the constants.

So that's another way to see why it simplifies to 14.

Therefore, I can be confident that the simplified form is 14.

**Final Answer**
The simplified expression is \boxed{14}.

To simplify the expression:

$$
(u + 4)(u - 1) - (u - 3)(u + 6)
$$

we proceed step by step by first expanding each product and then combining like terms.

---

### Step 1: Expand the first product

$$
(u + 4)(u - 1)
$$

Use the distributive property (FOIL method):

- First: $ u \cdot u = u^2 $
- Outer: $ u \cdot (-1) = -u $
- Inner: $ 4 \cdot u = 4u $
- Last: $ 4 \cdot (-1) = -4 $

Combine terms:

$$
u^2 - u + 4u - 4 = u^2 + 3u - 4
$$

---

### Step 2: Expand the second product

$$
(u - 3)(u + 6)
$$

Again, use the distributive property:

- First: $ u \cdot u = u^2 $
- Outer: $ u \cdot 6 = 6u $
- Inner: $ -3 \cdot u = -3u $
- Last: $ -3 \cdot 6 = -18 $

Combine terms:

$$
u^2 + 6u - 3u - 18 = u^2 + 3u - 18
$$

---

### Step 3: Subtract the second expression from the first

$$
(u^2 + 3u - 4) - (u^2 + 3u - 18)
$$

Distribute the negative sign across the second expression:

$$
u^2 + 3u - 4 - u^2 - 3u + 18
$$

Now combine like terms:

- $ u^2 - u^2 = 0 $
- $ 3u - 3u = 0 $
- $ -4 + 18 = 14 $

---

### Final Simplified Expression

$$
\boxed{14}
$$

[Assistant]
Explanation:
Okay, so I need to simplify this expression: (u + 4)(u - 1) minus (u - 3)(u + 6). Let me think about how to approach this. I remember that to simplify expressions like this, I should probably expand both products first and then combine like terms. Let me start with the first part, (u + 4)(u - 1).

Alright, expanding that. I think I need to use the distributive property, also known as the FOIL method for binomials. Let me recall: FOIL stands for First, Outer, Inner, Last. So, for (u + 4)(u - 1):

First terms: u * u = u²

Outer terms: u * (-1) = -u

Inner terms: 4 * u = 4u

Last terms: 4 * (-1) = -4

Now, adding all those together: u² - u + 4u - 4. Let me combine like terms here. The -u and +4u can be combined. That would be (-1u + 4u) which is 3u. So the first part simplifies to u² + 3u - 4. Let me check that again.

Wait, u² - u + 4u - 4. Yes, combining -u + 4u is 3u. So that's correct. So the first part is u² + 3u - 4. Okay, got that.

Now, moving on to the second part: (u - 3)(u + 6). Again, I need to expand this. Let me apply the same FOIL method.

First terms: u * u = u²

Outer terms: u * 6 = 6u

Inner terms: -3 * u = -3u

Last terms: -3 * 6 = -18

Adding those together: u² + 6u - 3u - 18. Combining like terms here: 6u - 3u is 3u. So this simplifies to u² + 3u - 18. Wait, that's interesting. Both the first and second parts have u² + 3u terms? Let me check again.

First part: (u + 4)(u - 1) = u² - u + 4u - 4 = u² + 3u - 4. Correct.

Second part: (u - 3)(u + 6) = u² + 6u - 3u - 18 = u² + 3u - 18. Yes, that's right. So both expanded forms have u² + 3u, but different constants.

Now, the original expression is the first part minus the second part. So that would be:

(u² + 3u - 4) - (u² + 3u - 18).

Wait, I need to be careful with the subtraction. The entire second part is being subtracted. So I need to distribute the negative sign to each term in the second polynomial. Let me write that out:

= u² + 3u - 4 - u² - 3u + 18.

Wait, let me check that again. If I have (A) - (B), where A is u² + 3u - 4 and B is u² + 3u - 18, then it's A - B = (u² + 3u - 4) - u² - 3u + 18. Wait, no. Wait, subtracting each term in B. So:

Original expression: (u² + 3u - 4) - (u² + 3u - 18)

Which is equal to u² + 3u - 4 - u² - 3u + 18. Because subtracting u² is -u², subtracting 3u is -3u, and subtracting (-18) is +18.

Now, let me combine like terms here. Let's see:

u² - u² = 0.

3u - 3u = 0.

Then, -4 + 18 = 14.

So all the terms cancel out except for 14? That seems surprising. Let me check my steps again to make sure I didn't make a mistake.

First expansion: (u + 4)(u - 1) = u² + 3u - 4. Correct.

Second expansion: (u - 3)(u + 6) = u² + 3u - 18. Correct.

Subtracting the second from the first: (u² + 3u - 4) - (u² + 3u - 18).

Distribute the negative sign: u² + 3u - 4 - u² - 3u + 18.

Combine like terms:

u² - u² = 0.

3u - 3u = 0.

-4 + 18 = 14.

So the entire expression simplifies to 14? That seems too simple. Let me try plugging in a value for u to check. Let's pick u = 0.

Original expression: (0 + 4)(0 - 1) - (0 - 3)(0 + 6) = (4)(-1) - (-3)(6) = -4 - (-18) = -4 + 18 = 14.

Okay, that works. Let me try another value, say u = 1.

Original expression: (1 + 4)(1 - 1) - (1 - 3)(1 + 6) = (5)(0) - (-2)(7) = 0 - (-14) = 14.

Hmm, also 14. Let me try u = 2.

Original expression: (2 + 4)(2 - 1) - (2 - 3)(2 + 6) = (6)(1) - (-1)(8) = 6 - (-8) = 6 + 8 = 14.

Same result. So it seems like regardless of the value of u, the expression simplifies to 14. Therefore, my simplification is correct.

But just to make sure, let me try expanding again step by step.

First part: (u + 4)(u - 1). Multiply term by term:

u*(u) + u*(-1) + 4*(u) + 4*(-1) = u² - u + 4u - 4 = u² + 3u - 4. Correct.

Second part: (u - 3)(u + 6). Multiply term by term:

u*(u) + u*(6) + (-3)*(u) + (-3)*(6) = u² + 6u - 3u - 18 = u² + 3u - 18. Correct.

Subtracting the second from the first:

(u² + 3u - 4) - (u² + 3u - 18) = u² + 3u - 4 - u² - 3u + 18.

Combine like terms:

u² - u² = 0.

3u - 3u = 0.

-4 + 18 = 14.

Yes, that's correct. So the simplified form is 14.

Therefore, the answer is 14.

But just to make sure there isn't a trick or something I missed. The problem says "simplify," and 14 is a constant, so that's as simplified as it gets.

Alternatively, maybe there's another way to approach this problem? Let me think. Maybe by combining the two expressions before expanding? Let me try that.

Original expression: (u + 4)(u - 1) - (u - 3)(u + 6).

Alternatively, maybe factor something out? But since they are both quadratic terms, perhaps not. Alternatively, maybe notice that both have u² + 3u terms? Wait, when I expanded them, both had u² + 3u, so when subtracting, they cancel.

So, if I think of it as [(u² + 3u) - 4] - [(u² + 3u) - 18] = (u² + 3u) - 4 - u² - 3u + 18 = (-4 + 18) = 14.

Yes, that's another way to see it. The u² and 3u terms cancel out, leaving just the constants.

So that's another way to see why it simplifies to 14.

Therefore, I can be confident that the simplified form is 14.

**Final Answer**
The simplified expression is \boxed{14}.

To simplify the expression:

$$
(u + 4)(u - 1) - (u - 3)(u + 6)
$$

we proceed step by step by first expanding each product and then combining like terms.

---

### Step 1: Expand the first product

$$
(u + 4)(u - 1)
$$

Use the distributive property (FOIL method):

- First: $ u \cdot u = u^2 $
- Outer: $ u \cdot (-1) = -u $
- Inner: $ 4 \cdot u = 4u $
- Last: $ 4 \cdot (-1) = -4 $

Combine terms:

$$
u^2 - u + 4u - 4 = u^2 + 3u - 4
$$

---

### Step 2: Expand the second product

$$
(u - 3)(u + 6)
$$

Again, use the distributive property:

- First: $ u \cdot u = u^2 $
- Outer: $ u \cdot 6 = 6u $
- Inner: $ -3 \cdot u = -3u $
- Last: $ -3 \cdot 6 = -18 $

Combine terms:

$$
u^2 + 6u - 3u - 18 = u^2 + 3u - 18
$$

---

### Step 3: Subtract the second expression from the first

$$
(u^2 + 3u - 4) - (u^2 + 3u - 18)
$$

Distribute the negative sign across the second expression:

$$
u^2 + 3u - 4 - u^2 - 3u + 18
$$

Now combine like terms:

- $ u^2 - u^2 = 0 $
- $ 3u - 3u = 0 $
- $ -4 + 18 = 14 $

---

### Final Simplified Expression

$$
\boxed{14}
$$

Solution: Expanding the first product, the distribute property shows that  $$(u+4)(u-1) = u^2 + 4u - u - 4 = u^2 + 3u - 4.$$The second product becomes $$(u-3)(u+6) = u^2 - 3u + 6u - 18 = u^2 + 3u - 18.$$Subtracting, both the $u^2$ and the $3u$ terms cancel, leaving an answer of $-4 - (-18) = \boxed{14}$.