Explanation:
Okay, so I need to figure out by what percentage the area of a square decreases when two opposite sides are decreased by 40% and the other two sides are increased by 50% to form a rectangle. Hmm, let me start by recalling some basics about squares and rectangles.

First, a square has all sides equal. Let me denote the length of each side of the square as 's'. So, the original area of the square would be s squared, right? So, Area_original = s².

Now, the problem says that two opposite sides are decreased by 40%, and the other two sides are increased by 50%. Wait, hold on. In a square, all sides are equal, so if we decrease two opposite sides by 40%, that would mean those sides become shorter, and the other two sides, which are opposite to them, are increased by 50%. But since all sides are the same length initially, decreasing two sides and increasing the other two would result in a rectangle where the lengths of the sides are different. Let me visualize this.

Let me think. Let's say the original square has sides of length 's'. If we take two opposite sides and decrease them by 40%, their new length would be s minus 40% of s. Which is s*(1 - 0.40) = s*0.60. Similarly, the other two sides are increased by 50%, so their new length is s + 50% of s = s*(1 + 0.50) = s*1.50.

Therefore, the resulting figure is a rectangle with length and width of 0.60s and 1.50s? Wait, but which sides are which? Let me make sure. Since it's a rectangle, opposite sides are equal. So, if two opposite sides are decreased by 40%, they become 0.6s, and the other two sides, which are opposite to them, are increased by 50%, so they become 1.5s. Therefore, the rectangle has sides of length 0.6s and 1.5s. Therefore, the area of the rectangle would be length multiplied by width, so 0.6s * 1.5s.

Let me compute that. 0.6 * 1.5 is... 0.9. So, Area_new = 0.9s². Wait, so the original area was s², and the new area is 0.9s². Therefore, the area has decreased by 0.1s², which is 10% of the original area. Therefore, the percentage decrease is 10%? Is that correct?

Wait, let me check again. Maybe I made a mistake in calculation. Let me recalculate 0.6 multiplied by 1.5. 0.6 * 1.5. Well, 0.6 * 1 is 0.6, and 0.6 * 0.5 is 0.3, so adding them together gives 0.6 + 0.3 = 0.9. Yes, that's correct. So 0.9s². Therefore, the area is decreased by 10%. So the answer is 10%?

But let me think again. Maybe I need to check with actual numbers. Let me take a specific example. Let me assume the original square has sides of length 10 units. Then, original area is 10*10 = 100 square units.

Now, decreasing two opposite sides by 40%: 40% of 10 is 4, so the new length is 10 - 4 = 6 units. The other two sides are increased by 50%: 50% of 10 is 5, so the new length is 10 + 5 = 15 units. Therefore, the rectangle has sides of 6 and 15. Therefore, the area is 6*15 = 90 square units. Original area was 100, new area is 90. So the decrease is 10 square units, which is 10% of the original area. So that's 10% decrease. So that seems to confirm the previous result.

Wait, so why does the problem say "the lengths of two opposite sides are decreased by 40% while the lengths of the other two sides are increased by 50% to form a rectangle." So, in this case, the rectangle has sides of 6 and 15. So, the area is 90, which is 90% of the original area. Therefore, the decrease is 10%. Therefore, the answer is 10%? But let me check again with another example to be sure.

Let me take a square with side length 20. Original area is 400. Decreasing two sides by 40%: 40% of 20 is 8, so new length is 12. Increasing the other two sides by 50%: 50% of 20 is 10, so new length is 30. Area of rectangle is 12*30 = 360. Original area was 400. The difference is 40, which is 10% of 400. So again, 10% decrease. So that seems consistent.

Therefore, maybe the answer is 10%? But let me think again if there's a different interpretation of the problem. Maybe the problem is not about changing two opposite sides and the other two sides, but maybe something else? Wait, the problem says "the lengths of two opposite sides of a square are decreased by 40% while the lengths of the other two sides are increased by 50% to form a rectangle." So, that seems straightforward. Since a square has four sides, two opposite sides are decreased, and the other two opposite sides are increased. Therefore, resulting in a rectangle with sides of 0.6s and 1.5s. Therefore, area is 0.6*1.5*s² = 0.9s². Therefore, 10% decrease. So, seems like 10% is the answer.

But let me check if there's another way to interpret the problem. For example, maybe the problem is referring to changing the lengths of two opposite sides by 40% and the other two sides by 50%, but not necessarily that the two opposite sides are decreased and the other two are increased. Wait, but the problem says "the lengths of two opposite sides... are decreased by 40% while the lengths of the other two sides are increased by 50%". So, the wording is clear: two opposite sides are decreased, the other two are increased. So, that's the same as before.

Alternatively, maybe someone could think that the sides are being changed in some other way? For example, maybe the problem is referring to changing the length of two opposite sides by 40% each, and the other two sides by 50% each? But that's the same as what I did. Because if you decrease two opposite sides by 40%, then their lengths become 0.6s, and the other two sides are increased by 50%, so they become 1.5s. So, the rectangle has sides of 0.6s and 1.5s. Therefore, area is 0.6*1.5*s² = 0.9s². So, 10% decrease.

Alternatively, maybe someone could think that the problem is changing the length of two opposite sides by 40% and the other two sides by 50%, but not necessarily that the two opposite sides are decreased and the other two are increased. Wait, but the problem says "the lengths of two opposite sides... are decreased by 40% while the lengths of the other two sides are increased by 50%". So, the wording is very clear. So, two sides are decreased, two sides are increased. Therefore, the area is 0.6s * 1.5s = 0.9s². Therefore, 10% decrease.

Alternatively, maybe the problem is referring to changing the length of two opposite sides by 40% and the other two sides by 50%, but not necessarily that the two opposite sides are decreased and the other two are increased. Wait, but the problem says "the lengths of two opposite sides... are decreased by 40% while the lengths of the other two sides are increased by 50%". So, the "while" here is indicating that the other two sides are increased. So, that's clear.

Alternatively, maybe someone could think that the problem is changing the length of two opposite sides by 40% and the other two sides by 50%, but not specifying which direction. But the problem says "decreased" and "increased", so that's clear.

Alternatively, maybe there's a mistake in assuming that the sides are being changed independently? Wait, but in a square, all sides are equal. So, if you decrease two opposite sides by 40%, they become shorter, and the other two sides are increased by 50%, so they become longer. Therefore, the resulting figure is a rectangle with sides of different lengths. Therefore, the area is as calculated.

Alternatively, maybe the problem is referring to changing the length of two opposite sides by 40% each, and the other two sides by 50% each, but in some other way? For example, maybe the problem is referring to changing the length of two adjacent sides? But no, the problem says "two opposite sides".

Alternatively, maybe the problem is referring to changing the length of two opposite sides by 40% and the other two sides by 50%, but in such a way that the sides are not just scaled? But no, the problem says "the lengths of two opposite sides... are decreased by 40%", which is a percentage decrease, so that would be scaling by 0.6, and the other sides are increased by 50%, scaling by 1.5.

Alternatively, maybe the problem is referring to changing the length of two opposite sides by 40% of the original length, and the other two sides by 50% of the original length? Which is the same as scaling. So, for example, if the original side is s, then decreasing two sides by 40% would mean subtracting 0.4s from each, so new length is s - 0.4s = 0.6s. Similarly, increasing the other two sides by 50% would mean adding 0.5s to each, so new length is s + 0.5s = 1.5s. Therefore, same as before.

Therefore, the area is 0.6s * 1.5s = 0.9s². Therefore, the area is decreased by 10%. Therefore, the answer is 10%.

But let me check once again with algebra. Let me denote the original side length as s. Original area is s². After the changes, the sides are 0.6s and 1.5s. Therefore, new area is 0.6s * 1.5s = 0.9s². Therefore, the decrease in area is s² - 0.9s² = 0.1s². Therefore, the percentage decrease is (0.1s² / s²) * 100% = 10%. So, that's 10%.

Therefore, the answer is 10%. Therefore, the square's area decreases by 10%.

But let me check if there's a different interpretation where the answer is different. For example, maybe the problem is referring to changing the length of two opposite sides by 40% and the other two sides by 50%, but not necessarily that the two opposite sides are decreased and the other two are increased. Wait, but the problem says "the lengths of two opposite sides... are decreased by 40% while the lengths of the other two sides are increased by 50%". So, the wording is very clear. So, two sides are decreased, two sides are increased. Therefore, the area is 0.6s * 1.5s = 0.9s². Therefore, 10% decrease.

Alternatively, maybe someone could think that the problem is referring to changing the length of two opposite sides by 40% and the other two sides by 50%, but in such a way that the sides are not just scaled? For example, maybe the problem is referring to changing the length of two opposite sides by 40% of the original length, and the other two sides by 50% of the original length, but in some other way? But that would be the same as scaling. For example, if you decrease two sides by 40% of their original length, that is equivalent to multiplying by 0.6. Similarly, increasing the other two sides by 50% of their original length is multiplying by 1.5. Therefore, same result.

Alternatively, maybe someone could think that the problem is referring to changing the length of two opposite sides by 40% of the original length, and the other two sides by 50% of the original length, but in the same direction? For example, decreasing two sides by 40% and increasing the other two sides by 50%? But that's what we already considered.

Alternatively, maybe someone could think that the problem is referring to changing the length of two opposite sides by 40% and the other two sides by 50%, but not specifying which sides are which? But the problem says "two opposite sides... are decreased by 40% while the lengths of the other two sides are increased by 50%", so that's clear.

Alternatively, maybe the problem is referring to changing the length of two opposite sides by 40% and the other two sides by 50%, but in such a way that the sides are not just scaled? For example, maybe the problem is referring to changing the length of two opposite sides by 40% of the other sides? But that would be a different interpretation, but the problem says "decreased by 40%", which is a percentage decrease relative to the original length.

Alternatively, maybe the problem is referring to changing the length of two opposite sides by 40% of the original length, and the other two sides by 50% of the original length, but in such a way that the sides are not just scaled? For example, maybe the problem is referring to changing the length of two opposite sides by 40% of the original length, so that they become s - 0.4s = 0.6s, and the other two sides are increased by 50% of the original length, so they become s + 0.5s = 1.5s. Which is the same as before.

Therefore, I think my original reasoning is correct. Therefore, the area decreases by 10%.

But let me check once again with another example. Let me take s = 1. Original area is 1. After changes, sides are 0.6 and 1.5. Area is 0.6 * 1.5 = 0.9. So, decrease is 0.1, which is 10%. So, same result.

Alternatively, maybe the problem is referring to changing the length of two opposite sides by 40% and the other two sides by 50%, but in a different way? For example, maybe the problem is referring to changing the length of two opposite sides by 40% of the other sides? But that would be a different problem. For example, if you have a square with sides of length s, and you decrease two opposite sides by 40% of the length of the other sides. But that would be different. However, the problem says "decreased by 40%", which is a percentage decrease relative to their original length, not relative to the other sides. Therefore, that interpretation is not correct.

Alternatively, maybe the problem is referring to changing the length of two opposite sides by 40% and the other two sides by 50%, but in such a way that the sides are not just scaled? For example, maybe the problem is referring to changing the length of two opposite sides by 40% of the original length, and the other two sides by 50% of the original length, but in some other way? But that's the same as scaling.

Alternatively, maybe the problem is referring to changing the length of two opposite sides by 40% and the other two sides by 50%, but in such a way that the sides are not just scaled? For example, maybe the problem is referring to changing the length of two opposite sides by 40% of the original length, and the other two sides by 50% of the original length, but in some other way? But again, that's the same as scaling.

Therefore, I think my original answer is correct. The area decreases by 10%. Therefore, the answer is 10%.

But let me check once again with algebra. Let me denote the original side length as s. Original area is s². After the changes, the sides are 0.6s and 1.5s. Therefore, new area is 0.6s * 1.5s = 0.9s². Therefore, the area is decreased by 0.1s², which is 10% of the original area. Therefore, the percentage decrease is 10%.

Therefore, I think the answer is 10%. Therefore, the square's area decreases by 10%.

But let me check if there's a different way to calculate percentage decrease. The formula for percentage decrease is [(Original - New)/Original] * 100%. So, [(s² - 0.9s²)/s²] * 100% = (0.1s²/s²)*100% = 10%. So, same result.

Therefore, I think the answer is 10%. Therefore, the square's area decreases by 10%.

But just to make sure, let me think if there's any chance that the problem is referring to something else. For example, maybe the problem is referring to changing the length of two opposite sides by 40% and the other two sides by 50%, but not in the way I thought. For example, maybe the problem is referring to changing the length of two opposite sides by 40% of the original length, and the other two sides by 50% of the original length, but in such a way that the sides are not just scaled? Wait, but that's the same as scaling. For example, decreasing by 40% is scaling by 0.6, increasing by 50% is scaling by 1.5. Therefore, the area is scaled by 0.6 * 1.5 = 0.9, so 90% of the original area, so 10% decrease.

Alternatively, maybe the problem is referring to changing the length of two opposite sides by 40% and the other two sides by 50%, but in such a way that the sides are not just scaled? For example, maybe the problem is referring to changing the length of two opposite sides by 40% of the other sides? But that would be different. For example, if you have a square with sides of length s, and you decrease two opposite sides by 40% of the length of the other sides. Wait, but the other sides are also s. So, decreasing two sides by 40% of s would be 0.4s, so new length is s - 0.4s = 0.6s. Similarly, increasing the other two sides by 50% of the length of the other sides, which is 0.5s, so new length is s + 0.5s = 1.5s. Therefore, same result. Therefore, same area.

Therefore, regardless of how I interpret it, the result is the same. Therefore, the answer is 10%.

Therefore, I think the answer is 10%. Therefore, the square's area decreases by 10%.

**Final Answer**
The square's area decreases by \boxed{10} percent.

To determine the percentage decrease in the area of a square after modifying its sides, we begin by analyzing the original and modified dimensions.

---

### Step 1: Define the Original Square

Let the side length of the square be $ s $.
Then, the **original area** of the square is:

$$
A_{\text{original}} = s^2
$$

---

### Step 2: Apply the Percentage Changes

- Two opposite sides are **decreased by 40%**, so their new length becomes:

$$
s - 0.40s = 0.60s
$$

- The other two opposite sides are **increased by 50%**, so their new length becomes:

$$
s + 0.50s = 1.50s
$$

Thus, the resulting figure is a **rectangle** with side lengths $ 0.60s $ and $ 1.50s $.

---

### Step 3: Calculate the New Area

The **new area** of the rectangle is:

$$
A_{\text{new}} = (0.60s)(1.50s) = 0.90s^2
$$

---

### Step 4: Determine the Percentage Decrease

The **decrease in area** is:

$$
\Delta A = A_{\text{original}} - A_{\text{new}} = s^2 - 0.90s^2 = 0.10s^2
$$

The **percentage decrease** is:

$$
\text{Percentage decrease} = \left( \frac{\Delta A}{A_{\text{original}}} \right) \times 100\% = \left( \frac{0.10s^2}{s^2} \right) \times 100\% = 10\%
$$

---

### Final Answer

$$
\boxed{10}
$$

[Assistant]
Explanation:
Okay, so I need to figure out by what percentage the area of a square decreases when two opposite sides are decreased by 40% and the other two sides are increased by 50% to form a rectangle. Hmm, let me start by recalling some basics about squares and rectangles.

First, a square has all sides equal. Let me denote the length of each side of the square as 's'. So, the original area of the square would be s squared, right? So, Area_original = s².

Now, the problem says that two opposite sides are decreased by 40%, and the other two sides are increased by 50%. Wait, hold on. In a square, all sides are equal, so if we decrease two opposite sides by 40%, that would mean those sides become shorter, and the other two sides, which are opposite to them, are increased by 50%. But since all sides are the same length initially, decreasing two sides and increasing the other two would result in a rectangle where the lengths of the sides are different. Let me visualize this.

Let me think. Let's say the original square has sides of length 's'. If we take two opposite sides and decrease them by 40%, their new length would be s minus 40% of s. Which is s*(1 - 0.40) = s*0.60. Similarly, the other two sides are increased by 50%, so their new length is s + 50% of s = s*(1 + 0.50) = s*1.50.

Therefore, the resulting figure is a rectangle with length and width of 0.60s and 1.50s? Wait, but which sides are which? Let me make sure. Since it's a rectangle, opposite sides are equal. So, if two opposite sides are decreased by 40%, they become 0.6s, and the other two sides, which are opposite to them, are increased by 50%, so they become 1.5s. Therefore, the rectangle has sides of length 0.6s and 1.5s. Therefore, the area of the rectangle would be length multiplied by width, so 0.6s * 1.5s.

Let me compute that. 0.6 * 1.5 is... 0.9. So, Area_new = 0.9s². Wait, so the original area was s², and the new area is 0.9s². Therefore, the area has decreased by 0.1s², which is 10% of the original area. Therefore, the percentage decrease is 10%? Is that correct?

Wait, let me check again. Maybe I made a mistake in calculation. Let me recalculate 0.6 multiplied by 1.5. 0.6 * 1.5. Well, 0.6 * 1 is 0.6, and 0.6 * 0.5 is 0.3, so adding them together gives 0.6 + 0.3 = 0.9. Yes, that's correct. So 0.9s². Therefore, the area is decreased by 10%. So the answer is 10%?

But let me think again. Maybe I need to check with actual numbers. Let me take a specific example. Let me assume the original square has sides of length 10 units. Then, original area is 10*10 = 100 square units.

Now, decreasing two opposite sides by 40%: 40% of 10 is 4, so the new length is 10 - 4 = 6 units. The other two sides are increased by 50%: 50% of 10 is 5, so the new length is 10 + 5 = 15 units. Therefore, the rectangle has sides of 6 and 15. Therefore, the area is 6*15 = 90 square units. Original area was 100, new area is 90. So the decrease is 10 square units, which is 10% of the original area. So that's 10% decrease. So that seems to confirm the previous result.

Wait, so why does the problem say "the lengths of two opposite sides are decreased by 40% while the lengths of the other two sides are increased by 50% to form a rectangle." So, in this case, the rectangle has sides of 6 and 15. So, the area is 90, which is 90% of the original area. Therefore, the decrease is 10%. Therefore, the answer is 10%? But let me check again with another example to be sure.

Let me take a square with side length 20. Original area is 400. Decreasing two sides by 40%: 40% of 20 is 8, so new length is 12. Increasing the other two sides by 50%: 50% of 20 is 10, so new length is 30. Area of rectangle is 12*30 = 360. Original area was 400. The difference is 40, which is 10% of 400. So again, 10% decrease. So that seems consistent.

Therefore, maybe the answer is 10%? But let me think again if there's a different interpretation of the problem. Maybe the problem is not about changing two opposite sides and the other two sides, but maybe something else? Wait, the problem says "the lengths of two opposite sides of a square are decreased by 40% while the lengths of the other two sides are increased by 50% to form a rectangle." So, that seems straightforward. Since a square has four sides, two opposite sides are decreased, and the other two opposite sides are increased. Therefore, resulting in a rectangle with sides of 0.6s and 1.5s. Therefore, area is 0.6*1.5*s² = 0.9s². Therefore, 10% decrease. So, seems like 10% is the answer.

But let me check if there's another way to interpret the problem. For example, maybe the problem is referring to changing the lengths of two opposite sides by 40% and the other two sides by 50%, but not necessarily that the two opposite sides are decreased and the other two are increased. Wait, but the problem says "the lengths of two opposite sides... are decreased by 40% while the lengths of the other two sides are increased by 50%". So, the wording is clear: two opposite sides are decreased, the other two are increased. So, that's the same as before.

Alternatively, maybe someone could think that the sides are being changed in some other way? For example, maybe the problem is referring to changing the length of two opposite sides by 40% each, and the other two sides by 50% each? But that's the same as what I did. Because if you decrease two opposite sides by 40%, then their lengths become 0.6s, and the other two sides are increased by 50%, so they become 1.5s. So, the rectangle has sides of 0.6s and 1.5s. Therefore, area is 0.6*1.5*s² = 0.9s². So, 10% decrease.

Alternatively, maybe someone could think that the problem is changing the length of two opposite sides by 40% and the other two sides by 50%, but not necessarily that the two opposite sides are decreased and the other two are increased. Wait, but the problem says "the lengths of two opposite sides... are decreased by 40% while the lengths of the other two sides are increased by 50%". So, the wording is very clear. So, two sides are decreased, two sides are increased. Therefore, the area is 0.6s * 1.5s = 0.9s². Therefore, 10% decrease.

Alternatively, maybe the problem is referring to changing the length of two opposite sides by 40% and the other two sides by 50%, but not necessarily that the two opposite sides are decreased and the other two are increased. Wait, but the problem says "the lengths of two opposite sides... are decreased by 40% while the lengths of the other two sides are increased by 50%". So, the "while" here is indicating that the other two sides are increased. So, that's clear.

Alternatively, maybe someone could think that the problem is changing the length of two opposite sides by 40% and the other two sides by 50%, but not specifying which direction. But the problem says "decreased" and "increased", so that's clear.

Alternatively, maybe there's a mistake in assuming that the sides are being changed independently? Wait, but in a square, all sides are equal. So, if you decrease two opposite sides by 40%, they become shorter, and the other two sides are increased by 50%, so they become longer. Therefore, the resulting figure is a rectangle with sides of different lengths. Therefore, the area is as calculated.

Alternatively, maybe the problem is referring to changing the length of two opposite sides by 40% each, and the other two sides by 50% each, but in some other way? For example, maybe the problem is referring to changing the length of two adjacent sides? But no, the problem says "two opposite sides".

Alternatively, maybe the problem is referring to changing the length of two opposite sides by 40% and the other two sides by 50%, but in such a way that the sides are not just scaled? But no, the problem says "the lengths of two opposite sides... are decreased by 40%", which is a percentage decrease, so that would be scaling by 0.6, and the other sides are increased by 50%, scaling by 1.5.

Alternatively, maybe the problem is referring to changing the length of two opposite sides by 40% of the original length, and the other two sides by 50% of the original length? Which is the same as scaling. So, for example, if the original side is s, then decreasing two sides by 40% would mean subtracting 0.4s from each, so new length is s - 0.4s = 0.6s. Similarly, increasing the other two sides by 50% would mean adding 0.5s to each, so new length is s + 0.5s = 1.5s. Therefore, same as before.

Therefore, the area is 0.6s * 1.5s = 0.9s². Therefore, the area is decreased by 10%. Therefore, the answer is 10%.

But let me check once again with algebra. Let me denote the original side length as s. Original area is s². After the changes, the sides are 0.6s and 1.5s. Therefore, new area is 0.6s * 1.5s = 0.9s². Therefore, the decrease in area is s² - 0.9s² = 0.1s². Therefore, the percentage decrease is (0.1s² / s²) * 100% = 10%. So, that's 10%.

Therefore, the answer is 10%. Therefore, the square's area decreases by 10%.

But let me check if there's a different interpretation where the answer is different. For example, maybe the problem is referring to changing the length of two opposite sides by 40% and the other two sides by 50%, but not necessarily that the two opposite sides are decreased and the other two are increased. Wait, but the problem says "the lengths of two opposite sides... are decreased by 40% while the lengths of the other two sides are increased by 50%". So, the wording is very clear. So, two sides are decreased, two sides are increased. Therefore, the area is 0.6s * 1.5s = 0.9s². Therefore, 10% decrease.

Alternatively, maybe someone could think that the problem is referring to changing the length of two opposite sides by 40% and the other two sides by 50%, but in such a way that the sides are not just scaled? For example, maybe the problem is referring to changing the length of two opposite sides by 40% of the original length, and the other two sides by 50% of the original length, but in some other way? But that would be the same as scaling. For example, if you decrease two sides by 40% of their original length, that is equivalent to multiplying by 0.6. Similarly, increasing the other two sides by 50% of their original length is multiplying by 1.5. Therefore, same result.

Alternatively, maybe someone could think that the problem is referring to changing the length of two opposite sides by 40% of the original length, and the other two sides by 50% of the original length, but in the same direction? For example, decreasing two sides by 40% and increasing the other two sides by 50%? But that's what we already considered.

Alternatively, maybe someone could think that the problem is referring to changing the length of two opposite sides by 40% and the other two sides by 50%, but not specifying which sides are which? But the problem says "two opposite sides... are decreased by 40% while the lengths of the other two sides are increased by 50%", so that's clear.

Alternatively, maybe the problem is referring to changing the length of two opposite sides by 40% and the other two sides by 50%, but in such a way that the sides are not just scaled? For example, maybe the problem is referring to changing the length of two opposite sides by 40% of the other sides? But that would be a different interpretation, but the problem says "decreased by 40%", which is a percentage decrease relative to the original length.

Alternatively, maybe the problem is referring to changing the length of two opposite sides by 40% of the original length, and the other two sides by 50% of the original length, but in such a way that the sides are not just scaled? For example, maybe the problem is referring to changing the length of two opposite sides by 40% of the original length, so that they become s - 0.4s = 0.6s, and the other two sides are increased by 50% of the original length, so they become s + 0.5s = 1.5s. Which is the same as before.

Therefore, I think my original reasoning is correct. Therefore, the area decreases by 10%.

But let me check once again with another example. Let me take s = 1. Original area is 1. After changes, sides are 0.6 and 1.5. Area is 0.6 * 1.5 = 0.9. So, decrease is 0.1, which is 10%. So, same result.

Alternatively, maybe the problem is referring to changing the length of two opposite sides by 40% and the other two sides by 50%, but in a different way? For example, maybe the problem is referring to changing the length of two opposite sides by 40% of the other sides? But that would be a different problem. For example, if you have a square with sides of length s, and you decrease two opposite sides by 40% of the length of the other sides. But that would be different. However, the problem says "decreased by 40%", which is a percentage decrease relative to their original length, not relative to the other sides. Therefore, that interpretation is not correct.

Alternatively, maybe the problem is referring to changing the length of two opposite sides by 40% and the other two sides by 50%, but in such a way that the sides are not just scaled? For example, maybe the problem is referring to changing the length of two opposite sides by 40% of the original length, and the other two sides by 50% of the original length, but in some other way? But that's the same as scaling.

Alternatively, maybe the problem is referring to changing the length of two opposite sides by 40% and the other two sides by 50%, but in such a way that the sides are not just scaled? For example, maybe the problem is referring to changing the length of two opposite sides by 40% of the original length, and the other two sides by 50% of the original length, but in some other way? But again, that's the same as scaling.

Therefore, I think my original answer is correct. The area decreases by 10%. Therefore, the answer is 10%.

But let me check once again with algebra. Let me denote the original side length as s. Original area is s². After the changes, the sides are 0.6s and 1.5s. Therefore, new area is 0.6s * 1.5s = 0.9s². Therefore, the area is decreased by 0.1s², which is 10% of the original area. Therefore, the percentage decrease is 10%.

Therefore, I think the answer is 10%. Therefore, the square's area decreases by 10%.

But let me check if there's a different way to calculate percentage decrease. The formula for percentage decrease is [(Original - New)/Original] * 100%. So, [(s² - 0.9s²)/s²] * 100% = (0.1s²/s²)*100% = 10%. So, same result.

Therefore, I think the answer is 10%. Therefore, the square's area decreases by 10%.

But just to make sure, let me think if there's any chance that the problem is referring to something else. For example, maybe the problem is referring to changing the length of two opposite sides by 40% and the other two sides by 50%, but not in the way I thought. For example, maybe the problem is referring to changing the length of two opposite sides by 40% of the original length, and the other two sides by 50% of the original length, but in such a way that the sides are not just scaled? Wait, but that's the same as scaling. For example, decreasing by 40% is scaling by 0.6, increasing by 50% is scaling by 1.5. Therefore, the area is scaled by 0.6 * 1.5 = 0.9, so 90% of the original area, so 10% decrease.

Alternatively, maybe the problem is referring to changing the length of two opposite sides by 40% and the other two sides by 50%, but in such a way that the sides are not just scaled? For example, maybe the problem is referring to changing the length of two opposite sides by 40% of the other sides? But that would be different. For example, if you have a square with sides of length s, and you decrease two opposite sides by 40% of the length of the other sides. Wait, but the other sides are also s. So, decreasing two sides by 40% of s would be 0.4s, so new length is s - 0.4s = 0.6s. Similarly, increasing the other two sides by 50% of the length of the other sides, which is 0.5s, so new length is s + 0.5s = 1.5s. Therefore, same result. Therefore, same area.

Therefore, regardless of how I interpret it, the result is the same. Therefore, the answer is 10%.

Therefore, I think the answer is 10%. Therefore, the square's area decreases by 10%.

**Final Answer**
The square's area decreases by \boxed{10} percent.

To determine the percentage decrease in the area of a square after modifying its sides, we begin by analyzing the original and modified dimensions.

---

### Step 1: Define the Original Square

Let the side length of the square be $ s $.
Then, the **original area** of the square is:

$$
A_{\text{original}} = s^2
$$

---

### Step 2: Apply the Percentage Changes

- Two opposite sides are **decreased by 40%**, so their new length becomes:

$$
s - 0.40s = 0.60s
$$

- The other two opposite sides are **increased by 50%**, so their new length becomes:

$$
s + 0.50s = 1.50s
$$

Thus, the resulting figure is a **rectangle** with side lengths $ 0.60s $ and $ 1.50s $.

---

### Step 3: Calculate the New Area

The **new area** of the rectangle is:

$$
A_{\text{new}} = (0.60s)(1.50s) = 0.90s^2
$$

---

### Step 4: Determine the Percentage Decrease

The **decrease in area** is:

$$
\Delta A = A_{\text{original}} - A_{\text{new}} = s^2 - 0.90s^2 = 0.10s^2
$$

The **percentage decrease** is:

$$
\text{Percentage decrease} = \left( \frac{\Delta A}{A_{\text{original}}} \right) \times 100\% = \left( \frac{0.10s^2}{s^2} \right) \times 100\% = 10\%
$$

---

### Final Answer

$$
\boxed{10}
$$

Solution: Let $A$ be the area of the square. The lengths of one pair of opposite sides was decreased by $40\%$, so the area became $.6A$. The other pair of sides were increased by $50\%$, so the area became $1.5\cdot .6 A = .9A$. Thus, the area decreased by $\boxed{10}$ percent.


Explanation:
Okay, so I need to solve this equation: tan 53° tan 81° tan x° = tan 53° + tan 81° + tan x°, where x is between 0 and 180 degrees. Hmm, let me think about how to approach this.

First, I remember that there are some trigonometric identities involving tangent functions. Maybe I can use the tangent addition formula or something related? Let me recall... The tangent addition formula is tan(A + B) = (tan A + tan B) / (1 - tan A tan B). But I'm not sure if that's directly applicable here.

Alternatively, maybe I can rearrange the given equation to make it look like something familiar. Let me write down the equation again:

tan 53° tan 81° tan x° = tan 53° + tan 81° + tan x°.

Let me denote tan 53° as T and tan 81° as S for simplicity. Then the equation becomes:

T * S * tan x = T + S + tan x.

Hmm, maybe I can rearrange terms to isolate tan x? Let me try moving all terms to one side:

T * S * tan x - tan x - T - S = 0.

Factor tan x from the first two terms:

tan x (T * S - 1) - (T + S) = 0.

Then, tan x (T * S - 1) = T + S.

Therefore, tan x = (T + S) / (T * S - 1).

So tan x equals (tan 53° + tan 81°) divided by (tan 53° tan 81° - 1).

Wait, this looks similar to the formula for tan(A + B). Let me recall that tan(A + B) = (tan A + tan B) / (1 - tan A tan B). But here, the denominator is tan A tan B - 1, which is negative of (1 - tan A tan B). So if I compare:

tan x = [tan A + tan B] / [tan A tan B - 1] = - [tan A + tan B] / [1 - tan A tan B] = - tan(A + B).

Therefore, tan x = - tan(A + B). Which implies that tan x = tan(- (A + B)). Therefore, x is equal to - (A + B) + k*180°, where k is an integer.

But since x is between 0 and 180°, we need to find the appropriate k. Let me check.

So A is 53°, B is 81°, so A + B = 53° + 81° = 134°. Therefore, tan x = - tan(134°).

But tan(-θ) = - tan θ, so tan x = tan(-134°). However, tan is periodic with period 180°, so tan(-134°) = tan(-134° + 180°) = tan(46°). Therefore, tan x = tan(46°).

Therefore, x could be 46° or 46° + 180°, but since x is between 0 and 180°, the only possible solution is 46°.

Wait, but let me verify this. Let me check if x = 46° satisfies the original equation.

First, compute tan 53°, tan 81°, tan 46°.

But maybe I can check using the identity. If tan x = - tan(134°), then tan x = tan(180° - 134°) = tan(46°). Because tan(180° - θ) = - tan θ. Wait, tan(180° - θ) = - tan θ. Therefore, tan(46°) = tan(180° - 134°) = tan(46°). Wait, maybe I need to think again.

Let me recall that tan(θ) = tan(θ + k*180°). So if tan x = - tan(134°), then tan x = tan(-134°). But tan(-134°) = tan(-134° + 180°) = tan(46°). Therefore, x is 46° or 46° + 180°, but since x is between 0 and 180°, x = 46°.

Alternatively, maybe there's another way. Let me check with actual values. Let me compute tan 53°, tan 81°, tan 46°, and see if the equation holds.

But I need to remember approximate values. Let me recall that tan 45° = 1, tan 60° ≈ 1.732, tan 30° ≈ 0.577.

But 53° is close to 53.13°, which is related to a 3-4-5 triangle. Wait, tan 53.13° is 4/3 ≈ 1.333. Similarly, tan 81° is a large value, since tan 60° is about 1.732, tan 81° is tan(90° - 9°) = cot 9° ≈ 6.3138.

Similarly, tan 46° is tan(45° + 1°) ≈ 1 + tan 1° / (1 - tan 1°) ??? Wait, maybe better to use calculator approximations.

But since I don't have a calculator here, maybe I can use exact relations? Wait, but maybe there's another identity.

Alternatively, since we derived that tan x = - tan(134°), which is tan(46°), so x = 46°, but let me check if that works.

Let me compute tan 53° tan 81° tan 46° and compare it with tan 53° + tan 81° + tan 46°.

But without exact values, maybe I can use the identity. Let me think.

Earlier, we had tan x = - tan(A + B) where A = 53°, B = 81°, so tan x = - tan(134°).

But tan(134°) = tan(180° - 46°) = - tan(46°). Therefore, tan x = - (- tan 46°) = tan 46°, so tan x = tan 46°, so x = 46° + k*180°, but since x is between 0 and 180°, x = 46°.

Therefore, the answer is 46°.

But let me check if there's another possible solution. For example, if tan x = tan 46°, then x could be 46° or 46° + 180°, but since x < 180°, only 46° is valid.

Alternatively, maybe there's another angle? Let me think. Suppose that tan x = tan(46°), so x = 46° or x = 46° + 180°, but since x is less than 180°, only 46°.

Alternatively, maybe there's another approach. Let me think again.

Suppose we have tan A tan B tan C = tan A + tan B + tan C.

Is there a known identity for this?

In general, if A + B + C = 180°, then tan A + tan B + tan C = tan A tan B tan C.

Wait, that's a known identity. If A + B + C = π (or 180°), then tan A + tan B + tan C = tan A tan B tan C.

So in this case, if we have tan 53° tan 81° tan x° = tan 53° + tan 81° + tan x°, then this would be equivalent to tan A + tan B + tan C = tan A tan B tan C, which implies that A + B + C = 180°, provided that none of the tangents are undefined.

Therefore, if 53° + 81° + x° = 180°, then x° = 180° - 53° - 81° = 180° - 134° = 46°.

Therefore, x = 46°.

So that's another way to see it.

Therefore, the answer is 46°.

But let me check if this identity is correct.

Yes, the identity is that if A + B + C = π, then tan A + tan B + tan C = tan A tan B tan C.

So if we have tan A + tan B + tan C = tan A tan B tan C, then it's necessary that A + B + C = π (assuming none of the tangents are undefined).

Therefore, in our problem, since tan 53° tan 81° tan x° = tan 53° + tan 81° + tan x°, then this implies that 53° + 81° + x° = 180°, so x = 46°.

Therefore, the answer is 46°.

But let me check if there are other possibilities. For example, if A + B + C = π + kπ, but since angles are in degrees between 0 and 180, and tan is periodic with period 180°, but if we have angles adding up to more than 180°, but since x is between 0 and 180°, then 53 + 81 + x would be between 53 + 81 + 0 = 134° and 53 + 81 + 180 = 314°. So if A + B + C = π + kπ, then possible k values?

But if k = 0, then A + B + C = π (180°). If k = 1, then A + B + C = 5π/2? Wait, no. Wait, the identity is that if A + B + C = π, then tan A + tan B + tan C = tan A tan B tan C. However, if A + B + C = π + kπ, then tan A + tan B + tan C = tan A tan B tan C? Let me check for k = 1.

Suppose A + B + C = π + π = 2π. Then tan A + tan B + tan C = tan A tan B tan C? Let me take A = B = C = 2π/3. Then tan(2π/3) = tan(120°) = -√3. So tan A + tan B + tan C = 3*(-√3) = -3√3. tan A tan B tan C = (-√3)^3 = -3√3. So equality holds.

But in this case, A + B + C = 2π. So the identity holds for A + B + C = π + kπ. However, in our problem, the equation tan A tan B tan C = tan A + tan B + tan C is equivalent to A + B + C = π + kπ.

But since angles are between 0 and 180°, let's check possible k.

If k = 0: A + B + C = π (180°).

If k = 1: A + B + C = π + π = 2π (360°). But since each angle is between 0 and 180°, the maximum sum would be 180° + 180° + 180° = 540°, but for k = 1, 2π = 360°, so possible? Let me check if there's a solution where 53 + 81 + x = 360°, so x = 360 - 53 - 81 = 226°, but x must be less than 180°, so no. Similarly, for k = -1, A + B + C = π - π = 0°, which is impossible since angles are positive.

Therefore, the only possible solution is k = 0, so A + B + C = 180°, so x = 46°.

Therefore, the answer is 46°.

But let me check with approximate values to be sure. Let me try to compute tan 53°, tan 81°, tan 46°, and see if the equation holds.

First, approximate values:

tan 53° ≈ 1.3270 (since tan 53° is approximately 1.327)

tan 81° ≈ 6.3138 (since tan 81° is about 6.3138)

tan 46° ≈ 1.0355 (since tan 45° = 1, tan 46° is slightly more)

Now compute left side: tan53 * tan81 * tan46 ≈ 1.3270 * 6.3138 * 1.0355

First compute 1.3270 * 6.3138 ≈ Let me calculate:

1.327 * 6.3138 ≈ 1.327 * 6 + 1.327 * 0.3138 ≈ 7.962 + 0.416 ≈ 8.378

Then multiply by 1.0355: 8.378 * 1.0355 ≈ 8.378 + 8.378 * 0.0355 ≈ 8.378 + 0.297 ≈ 8.675

Now compute the right side: tan53 + tan81 + tan46 ≈ 1.327 + 6.3138 + 1.0355 ≈ 1.327 + 6.3138 = 7.6408 + 1.0355 ≈ 8.6763

Wow, that's very close! So left side ≈ 8.675 and right side ≈ 8.6763. The slight difference is due to my approximate values. Therefore, this confirms that x = 46° is indeed the solution.

Therefore, the answer is 46 degrees.

But just to make sure there are no other solutions, let me think. Suppose there is another angle x where tan x = tan 46°, but in the range 0 < x < 180°, tan x is positive in first and third quadrants. However, since x is between 0 and 180°, third quadrant would be between 180° and 270°, but x is less than 180°, so only first quadrant. Therefore, the only solution is 46°.

Alternatively, if there was another angle where tan x equals something else, but according to our previous derivation, the equation reduces to x = 46°, so that's unique.

Therefore, I can be confident that the answer is 46°.

**Final Answer**
\boxed{46}

To solve the equation:

$$
\tan 53^\circ \tan 81^\circ \tan x^\circ = \tan 53^\circ + \tan 81^\circ + \tan x^\circ
$$

we begin by recognizing a key trigonometric identity. Specifically, for any three angles $ A, B, C $ such that:

$$
A + B + C = 180^\circ
$$

the following identity holds:

$$
\tan A + \tan B + \tan C = \tan A \tan B \tan C
$$

This is a well-known identity in trigonometry, and it is valid when none of the tangents are undefined (i.e., none of the angles are $ 90^\circ $ or $ 270^\circ $, etc.).

---

### Step 1: Apply the Identity

Let us denote:

- $ A = 53^\circ $
- $ B = 81^\circ $
- $ C = x^\circ $

Then the given equation becomes:

$$
\tan A \tan B \tan C = \tan A + \tan B + \tan C
$$

This matches the identity exactly, so we can conclude that:

$$
A + B + C = 180^\circ
$$

Substituting the known values:

$$
53^\circ + 81^\circ + x^\circ = 180^\circ
$$

Solving for $ x $:

$$
x = 180^\circ - 53^\circ - 81^\circ = 46^\circ
$$

---

### Step 2: Verify the Solution

To ensure that this is the correct solution, we can verify that the identity holds for $ x = 46^\circ $. That is, we check whether:

$$
\tan 53^\circ \tan 81^\circ \tan 46^\circ = \tan 53^\circ + \tan 81^\circ + \tan 46^\circ
$$

Using approximate values:

- $ \tan 53^\circ \approx 1.3270 $
- $ \tan 81^\circ \approx 6.3138 $
- $ \tan 46^\circ \approx 1.0355 $

Compute the left-hand side:

$$
1.3270 \times 6.3138 \times 1.0355 \approx 8.675
$$

Compute the right-hand side:

$$
1.3270 + 6.3138 + 1.0355 \approx 8.6763
$$

The values are extremely close, confirming the identity holds and that $ x = 46^\circ $ is indeed the correct solution.

---

### Step 3: Consider Other Possible Solutions

The tangent function is periodic with period $ 180^\circ $, and $ \tan x = \tan y $ implies $ x = y + k \cdot 180^\circ $ for some integer $ k $. However, since the problem restricts $ x $ to the interval $ 0 < x < 180^\circ $, the only valid solution is:

$$
x = 46^\circ
$$

---

### Final Answer

$$
\boxed{46}
$$

[Assistant]
Explanation:
Okay, so I need to solve this equation: tan 53° tan 81° tan x° = tan 53° + tan 81° + tan x°, where x is between 0 and 180 degrees. Hmm, let me think about how to approach this.

First, I remember that there are some trigonometric identities involving tangent functions. Maybe I can use the tangent addition formula or something related? Let me recall... The tangent addition formula is tan(A + B) = (tan A + tan B) / (1 - tan A tan B). But I'm not sure if that's directly applicable here.

Alternatively, maybe I can rearrange the given equation to make it look like something familiar. Let me write down the equation again:

tan 53° tan 81° tan x° = tan 53° + tan 81° + tan x°.

Let me denote tan 53° as T and tan 81° as S for simplicity. Then the equation becomes:

T * S * tan x = T + S + tan x.

Hmm, maybe I can rearrange terms to isolate tan x? Let me try moving all terms to one side:

T * S * tan x - tan x - T - S = 0.

Factor tan x from the first two terms:

tan x (T * S - 1) - (T + S) = 0.

Then, tan x (T * S - 1) = T + S.

Therefore, tan x = (T + S) / (T * S - 1).

So tan x equals (tan 53° + tan 81°) divided by (tan 53° tan 81° - 1).

Wait, this looks similar to the formula for tan(A + B). Let me recall that tan(A + B) = (tan A + tan B) / (1 - tan A tan B). But here, the denominator is tan A tan B - 1, which is negative of (1 - tan A tan B). So if I compare:

tan x = [tan A + tan B] / [tan A tan B - 1] = - [tan A + tan B] / [1 - tan A tan B] = - tan(A + B).

Therefore, tan x = - tan(A + B). Which implies that tan x = tan(- (A + B)). Therefore, x is equal to - (A + B) + k*180°, where k is an integer.

But since x is between 0 and 180°, we need to find the appropriate k. Let me check.

So A is 53°, B is 81°, so A + B = 53° + 81° = 134°. Therefore, tan x = - tan(134°).

But tan(-θ) = - tan θ, so tan x = tan(-134°). However, tan is periodic with period 180°, so tan(-134°) = tan(-134° + 180°) = tan(46°). Therefore, tan x = tan(46°).

Therefore, x could be 46° or 46° + 180°, but since x is between 0 and 180°, the only possible solution is 46°.

Wait, but let me verify this. Let me check if x = 46° satisfies the original equation.

First, compute tan 53°, tan 81°, tan 46°.

But maybe I can check using the identity. If tan x = - tan(134°), then tan x = tan(180° - 134°) = tan(46°). Because tan(180° - θ) = - tan θ. Wait, tan(180° - θ) = - tan θ. Therefore, tan(46°) = tan(180° - 134°) = tan(46°). Wait, maybe I need to think again.

Let me recall that tan(θ) = tan(θ + k*180°). So if tan x = - tan(134°), then tan x = tan(-134°). But tan(-134°) = tan(-134° + 180°) = tan(46°). Therefore, x is 46° or 46° + 180°, but since x is between 0 and 180°, x = 46°.

Alternatively, maybe there's another way. Let me check with actual values. Let me compute tan 53°, tan 81°, tan 46°, and see if the equation holds.

But I need to remember approximate values. Let me recall that tan 45° = 1, tan 60° ≈ 1.732, tan 30° ≈ 0.577.

But 53° is close to 53.13°, which is related to a 3-4-5 triangle. Wait, tan 53.13° is 4/3 ≈ 1.333. Similarly, tan 81° is a large value, since tan 60° is about 1.732, tan 81° is tan(90° - 9°) = cot 9° ≈ 6.3138.

Similarly, tan 46° is tan(45° + 1°) ≈ 1 + tan 1° / (1 - tan 1°) ??? Wait, maybe better to use calculator approximations.

But since I don't have a calculator here, maybe I can use exact relations? Wait, but maybe there's another identity.

Alternatively, since we derived that tan x = - tan(134°), which is tan(46°), so x = 46°, but let me check if that works.

Let me compute tan 53° tan 81° tan 46° and compare it with tan 53° + tan 81° + tan 46°.

But without exact values, maybe I can use the identity. Let me think.

Earlier, we had tan x = - tan(A + B) where A = 53°, B = 81°, so tan x = - tan(134°).

But tan(134°) = tan(180° - 46°) = - tan(46°). Therefore, tan x = - (- tan 46°) = tan 46°, so tan x = tan 46°, so x = 46° + k*180°, but since x is between 0 and 180°, x = 46°.

Therefore, the answer is 46°.

But let me check if there's another possible solution. For example, if tan x = tan 46°, then x could be 46° or 46° + 180°, but since x < 180°, only 46° is valid.

Alternatively, maybe there's another angle? Let me think. Suppose that tan x = tan(46°), so x = 46° or x = 46° + 180°, but since x is less than 180°, only 46°.

Alternatively, maybe there's another approach. Let me think again.

Suppose we have tan A tan B tan C = tan A + tan B + tan C.

Is there a known identity for this?

In general, if A + B + C = 180°, then tan A + tan B + tan C = tan A tan B tan C.

Wait, that's a known identity. If A + B + C = π (or 180°), then tan A + tan B + tan C = tan A tan B tan C.

So in this case, if we have tan 53° tan 81° tan x° = tan 53° + tan 81° + tan x°, then this would be equivalent to tan A + tan B + tan C = tan A tan B tan C, which implies that A + B + C = 180°, provided that none of the tangents are undefined.

Therefore, if 53° + 81° + x° = 180°, then x° = 180° - 53° - 81° = 180° - 134° = 46°.

Therefore, x = 46°.

So that's another way to see it.

Therefore, the answer is 46°.

But let me check if this identity is correct.

Yes, the identity is that if A + B + C = π, then tan A + tan B + tan C = tan A tan B tan C.

So if we have tan A + tan B + tan C = tan A tan B tan C, then it's necessary that A + B + C = π (assuming none of the tangents are undefined).

Therefore, in our problem, since tan 53° tan 81° tan x° = tan 53° + tan 81° + tan x°, then this implies that 53° + 81° + x° = 180°, so x = 46°.

Therefore, the answer is 46°.

But let me check if there are other possibilities. For example, if A + B + C = π + kπ, but since angles are in degrees between 0 and 180, and tan is periodic with period 180°, but if we have angles adding up to more than 180°, but since x is between 0 and 180°, then 53 + 81 + x would be between 53 + 81 + 0 = 134° and 53 + 81 + 180 = 314°. So if A + B + C = π + kπ, then possible k values?

But if k = 0, then A + B + C = π (180°). If k = 1, then A + B + C = 5π/2? Wait, no. Wait, the identity is that if A + B + C = π, then tan A + tan B + tan C = tan A tan B tan C. However, if A + B + C = π + kπ, then tan A + tan B + tan C = tan A tan B tan C? Let me check for k = 1.

Suppose A + B + C = π + π = 2π. Then tan A + tan B + tan C = tan A tan B tan C? Let me take A = B = C = 2π/3. Then tan(2π/3) = tan(120°) = -√3. So tan A + tan B + tan C = 3*(-√3) = -3√3. tan A tan B tan C = (-√3)^3 = -3√3. So equality holds.

But in this case, A + B + C = 2π. So the identity holds for A + B + C = π + kπ. However, in our problem, the equation tan A tan B tan C = tan A + tan B + tan C is equivalent to A + B + C = π + kπ.

But since angles are between 0 and 180°, let's check possible k.

If k = 0: A + B + C = π (180°).

If k = 1: A + B + C = π + π = 2π (360°). But since each angle is between 0 and 180°, the maximum sum would be 180° + 180° + 180° = 540°, but for k = 1, 2π = 360°, so possible? Let me check if there's a solution where 53 + 81 + x = 360°, so x = 360 - 53 - 81 = 226°, but x must be less than 180°, so no. Similarly, for k = -1, A + B + C = π - π = 0°, which is impossible since angles are positive.

Therefore, the only possible solution is k = 0, so A + B + C = 180°, so x = 46°.

Therefore, the answer is 46°.

But let me check with approximate values to be sure. Let me try to compute tan 53°, tan 81°, tan 46°, and see if the equation holds.

First, approximate values:

tan 53° ≈ 1.3270 (since tan 53° is approximately 1.327)

tan 81° ≈ 6.3138 (since tan 81° is about 6.3138)

tan 46° ≈ 1.0355 (since tan 45° = 1, tan 46° is slightly more)

Now compute left side: tan53 * tan81 * tan46 ≈ 1.3270 * 6.3138 * 1.0355

First compute 1.3270 * 6.3138 ≈ Let me calculate:

1.327 * 6.3138 ≈ 1.327 * 6 + 1.327 * 0.3138 ≈ 7.962 + 0.416 ≈ 8.378

Then multiply by 1.0355: 8.378 * 1.0355 ≈ 8.378 + 8.378 * 0.0355 ≈ 8.378 + 0.297 ≈ 8.675

Now compute the right side: tan53 + tan81 + tan46 ≈ 1.327 + 6.3138 + 1.0355 ≈ 1.327 + 6.3138 = 7.6408 + 1.0355 ≈ 8.6763

Wow, that's very close! So left side ≈ 8.675 and right side ≈ 8.6763. The slight difference is due to my approximate values. Therefore, this confirms that x = 46° is indeed the solution.

Therefore, the answer is 46 degrees.

But just to make sure there are no other solutions, let me think. Suppose there is another angle x where tan x = tan 46°, but in the range 0 < x < 180°, tan x is positive in first and third quadrants. However, since x is between 0 and 180°, third quadrant would be between 180° and 270°, but x is less than 180°, so only first quadrant. Therefore, the only solution is 46°.

Alternatively, if there was another angle where tan x equals something else, but according to our previous derivation, the equation reduces to x = 46°, so that's unique.

Therefore, I can be confident that the answer is 46°.

**Final Answer**
\boxed{46}

To solve the equation:

$$
\tan 53^\circ \tan 81^\circ \tan x^\circ = \tan 53^\circ + \tan 81^\circ + \tan x^\circ
$$

we begin by recognizing a key trigonometric identity. Specifically, for any three angles $ A, B, C $ such that:

$$
A + B + C = 180^\circ
$$

the following identity holds:

$$
\tan A + \tan B + \tan C = \tan A \tan B \tan C
$$

This is a well-known identity in trigonometry, and it is valid when none of the tangents are undefined (i.e., none of the angles are $ 90^\circ $ or $ 270^\circ $, etc.).

---

### Step 1: Apply the Identity

Let us denote:

- $ A = 53^\circ $
- $ B = 81^\circ $
- $ C = x^\circ $

Then the given equation becomes:

$$
\tan A \tan B \tan C = \tan A + \tan B + \tan C
$$

This matches the identity exactly, so we can conclude that:

$$
A + B + C = 180^\circ
$$

Substituting the known values:

$$
53^\circ + 81^\circ + x^\circ = 180^\circ
$$

Solving for $ x $:

$$
x = 180^\circ - 53^\circ - 81^\circ = 46^\circ
$$

---

### Step 2: Verify the Solution

To ensure that this is the correct solution, we can verify that the identity holds for $ x = 46^\circ $. That is, we check whether:

$$
\tan 53^\circ \tan 81^\circ \tan 46^\circ = \tan 53^\circ + \tan 81^\circ + \tan 46^\circ
$$

Using approximate values:

- $ \tan 53^\circ \approx 1.3270 $
- $ \tan 81^\circ \approx 6.3138 $
- $ \tan 46^\circ \approx 1.0355 $

Compute the left-hand side:

$$
1.3270 \times 6.3138 \times 1.0355 \approx 8.675
$$

Compute the right-hand side:

$$
1.3270 + 6.3138 + 1.0355 \approx 8.6763
$$

The values are extremely close, confirming the identity holds and that $ x = 46^\circ $ is indeed the correct solution.

---

### Step 3: Consider Other Possible Solutions

The tangent function is periodic with period $ 180^\circ $, and $ \tan x = \tan y $ implies $ x = y + k \cdot 180^\circ $ for some integer $ k $. However, since the problem restricts $ x $ to the interval $ 0 < x < 180^\circ $, the only valid solution is:

$$
x = 46^\circ
$$

---

### Final Answer

$$
\boxed{46}
$$

Solution:
Isolating $\tan x^\circ,$ we find
\begin{align*}
\tan x &= \frac{\tan 53^\circ + \tan 81^\circ}{\tan 53^\circ \tan 81^\circ - 1} \\
&= -\frac{\tan 53^\circ + \tan 81^\circ}{1 - \tan 53^\circ \tan 81^\circ}.
\end{align*}From the angle addition formula, this is equal to
\[-\tan (53^\circ + 81^\circ) = -\tan 134^\circ = \tan 46^\circ.\]Therefore, $x = \boxed{46}.$


Explanation:
Okay, so I need to compute the value of $ z + \frac{1}{z} + z^2 + \frac{1}{z^2} $ where $ z $ is a complex number satisfying $ z^5 = 1 $ and $ z \neq 1 $. Let me think about how to approach this.

First, since $ z^5 = 1 $, $ z $ is a 5th root of unity. The 5th roots of unity are the solutions to the equation $ z^5 - 1 = 0 $. These can be expressed in the form $ e^{2\pi i k/5} $ for $ k = 0, 1, 2, 3, 4 $. But since $ z \neq 1 $, we know that $ z $ is one of the primitive 5th roots of unity, meaning $ k = 1, 2, 3, 4 $.

Now, the problem is asking for $ z + \frac{1}{z} + z^2 + \frac{1}{z^2} $. Let me note that $ \frac{1}{z} $ is the complex conjugate of $ z $ if $ z $ lies on the unit circle, which it does because all roots of unity lie on the unit circle. Wait, but since $ z^5 = 1 $, then $ |z| = 1 $, so indeed $ \frac{1}{z} = \overline{z} $. However, I don't know if that's immediately helpful here. Maybe I can express $ \frac{1}{z} $ as $ z^4 $, since $ z^5 = 1 $ implies $ z^5 - 1 = 0 $, so $ z^5 = 1 $, so $ z^4 = \frac{1}{z} $. Similarly, $ \frac{1}{z^2} = z^{3} $. Let me check that:

Since $ z^5 = 1 $, multiplying both sides by $ z^{-2} $ gives $ z^{3} = z^{-2} $, so $ \frac{1}{z^2} = z^{3} $. Similarly, $ \frac{1}{z} = z^4 $. Therefore, the original expression can be rewritten as:

$ z + z^4 + z^2 + z^3 $.

So the problem reduces to computing $ z + z^2 + z^3 + z^4 $.

But wait, since $ z^5 = 1 $, the sum of all 5th roots of unity is 0. That is, $ 1 + z + z^2 + z^3 + z^4 = 0 $. Therefore, $ z + z^2 + z^3 + z^4 = -1 $. Therefore, the original expression is equal to -1? Wait, but let me check again.

Wait, the original expression was $ z + \frac{1}{z} + z^2 + \frac{1}{z^2} $, which we converted to $ z + z^4 + z^2 + z^3 $, which is indeed $ z + z^2 + z^3 + z^4 $, and since the sum of all roots is 0, that sum is -1. Therefore, the answer is -1? But let me verify this with an example.

Let me take a specific 5th root of unity and compute the value. Let's take $ z = e^{2\pi i /5} $. Then compute $ z + 1/z + z^2 + 1/z^2 $.

First, $ 1/z = e^{-2\pi i /5} $, and $ 1/z^2 = e^{-4\pi i /5} $. So the expression becomes:

$ e^{2\pi i /5} + e^{-2\pi i /5} + e^{4\pi i /5} + e^{-4\pi i /5} $.

These are the real parts of the complex numbers. Let me compute each term:

$ e^{2\pi i /5} + e^{-2\pi i /5} = 2\cos(2\pi/5) $

Similarly, $ e^{4\pi i /5} + e^{-4\pi i /5} = 2\cos(4\pi/5) $

Therefore, the total expression is $ 2\cos(2\pi/5) + 2\cos(4\pi/5) $.

Now, let me compute this numerically. Let me recall that $ \cos(4\pi/5) = \cos(\pi - \pi/5) = -\cos(\pi/5) $, and $ \cos(2\pi/5) $ is just $ \cos(2\pi/5) $. Therefore, the expression becomes:

$ 2\cos(2\pi/5) - 2\cos(\pi/5) $.

Hmm, so I need to compute this value. Let me recall exact values for these cosines. Alternatively, maybe I can relate them to known quantities.

Alternatively, perhaps I can use the identity that the sum $ z + z^2 + z^3 + z^4 = -1 $, so according to that, the answer is -1. But according to the numerical calculation, it's $ 2\cos(2\pi/5) + 2\cos(4\pi/5) $. Let me compute this numerically.

First, compute $ \cos(2\pi/5) $. Let me recall that $ 2\pi/5 $ is 72 degrees. The cosine of 72 degrees is approximately 0.3090. Similarly, $ \cos(4\pi/5) $ is $ \cos(144^\circ) $, which is approximately -0.8090. Therefore, substituting:

$ 2*(0.3090) + 2*(-0.8090) = 0.618 - 1.618 = -1.0 $. Oh! So that gives -1. So that matches the previous conclusion. Therefore, the answer is indeed -1. Therefore, the original expression is equal to -1.

But let me check again. Wait, so according to the sum of roots, since $ 1 + z + z^2 + z^3 + z^4 = 0 $, so $ z + z^2 + z^3 + z^4 = -1 $. Therefore, the expression is -1. Therefore, the answer is \boxed{-1}?

But let me think again. Wait, the problem says "Compute $ z + \frac{1}{z} + z^2 + \frac{1}{z^2} $", which we converted to $ z + z^4 + z^2 + z^3 $, which is indeed the sum of all roots except 1, so that sum is -1. Therefore, the answer is -1. Therefore, the answer is \boxed{-1}.

But let me check again with another approach. Maybe using algebra.

Since $ z^5 = 1 $, so $ z \neq 1 $, so $ z^4 + z^3 + z^2 + z + 1 = 0 $, so $ z^4 + z^3 + z^2 + z = -1 $. Therefore, the expression $ z + z^2 + z^3 + z^4 = -1 $. Therefore, the answer is -1. Therefore, regardless of which root we take, the value is -1. Therefore, the answer is \boxed{-1}.

But let me check if there's another way to compute this expression. For example, maybe by squaring or something else. Let me try.

Let me denote $ S = z + \frac{1}{z} + z^2 + \frac{1}{z^2} $. Let me note that $ z + \frac{1}{z} = z + z^4 $, and $ z^2 + \frac{1}{z^2} = z^2 + z^3 $. Therefore, S is equal to $ (z + z^4) + (z^2 + z^3) $, which is again the same as before. Alternatively, maybe I can compute $ (z + 1/z)^2 $ or something like that.

Let me compute $ (z + 1/z)^2 = z^2 + 2 + 1/z^2 $. Therefore, $ z^2 + 1/z^2 = (z + 1/z)^2 - 2 $. Similarly, if I let $ A = z + 1/z $, then $ A^2 = z^2 + 2 + 1/z^2 $, so $ z^2 + 1/z^2 = A^2 - 2 $. Therefore, the original expression S is $ A + (A^2 - 2) $. Wait, no. Wait, S is $ z + 1/z + z^2 + 1/z^2 $, which is $ A + (z^2 + 1/z^2) $. Which is $ A + (A^2 - 2) $. Therefore, S = A + A^2 - 2. But then I need to find A.

Alternatively, maybe compute A first. Let me compute $ A = z + 1/z = z + z^4 $. Let me compute $ A $. Let me note that $ z^5 = 1 $, so $ z \neq 0 $, so we can write $ A = z + z^4 $. Similarly, since $ z^4 = z^{-1} $, so $ A = z + z^{-1} $. Let me compute $ A $.

Alternatively, since $ z^5 = 1 $, then $ z^5 - 1 = 0 $, which factors as $ (z - 1)(z^4 + z^3 + z^2 + z + 1) = 0 $. Since $ z \neq 1 $, then $ z^4 + z^3 + z^2 + z + 1 = 0 $, so $ z^4 + z^3 + z^2 + z = -1 $, which is the same as before.

Alternatively, maybe compute $ A = z + z^4 $. Let me compute $ A + z^2 + z^3 = -1 $. But that's the same as before.

Alternatively, maybe compute $ A = z + z^4 $. Let me compute $ A $:

Let me note that $ z^4 = \overline{z} $, since $ |z| = 1 $, so $ \overline{z} = 1/z $. Therefore, $ A = z + \overline{z} = 2\Re(z) $. Similarly, $ z^2 + z^3 = z^2 + z^{-2} = 2\Re(z^2) $. Therefore, S = 2\Re(z) + 2\Re(z^2). But maybe this is more complicated.

Alternatively, since we already have that S = -1, maybe that's the simplest way. But let me check if there's another way to compute S.

Alternatively, let me compute $ S = z + z^2 + z^3 + z^4 = -1 $, so that's straightforward. Therefore, the answer is -1. Therefore, I think that's the answer.

But let me check again with another example. Let me take $ z = e^{2\pi i /5} $, compute $ z + 1/z + z^2 + 1/z^2 $.

As before, $ z = e^{2\pi i /5} $, so $ 1/z = e^{-2\pi i /5} $, $ z^2 = e^{4\pi i /5} $, $ 1/z^2 = e^{-4\pi i /5} $. Therefore, adding them up:

$ e^{2\pi i /5} + e^{-2\pi i /5} + e^{4\pi i /5} + e^{-4\pi i /5} $.

Which is $ 2\cos(2\pi/5) + 2\cos(4\pi/5) $. As before, which is $ 2\cos(72^\circ) + 2\cos(144^\circ) $.

Now, let me compute these values numerically:

First, $ \cos(72^\circ) \approx 0.3090 $, so 2*0.3090 ≈ 0.618.

Then, $ \cos(144^\circ) = \cos(180^\circ - 36^\circ) = -\cos(36^\circ) ≈ -0.8090 $, so 2*(-0.8090) ≈ -1.618.

Adding them together: 0.618 - 1.618 = -1.0. So that's exactly -1. Therefore, this numerical example confirms that the value is indeed -1.

Therefore, regardless of the approach, the answer is -1. Therefore, I think the answer is \boxed{-1}.

But just to make sure, let me think if there's any possible mistake here. For example, could there be a case where $ z + 1/z + z^2 + 1/z^2 $ is different? But since all the non-1 roots satisfy the same equation, and the sum is always -1, regardless of which root we take. For example, take $ z = e^{4\pi i /5} $, which is another root. Then $ 1/z = e^{-4\pi i /5} $, and $ z^2 = e^{8\pi i /5} = e^{-2\pi i /5} $, and $ 1/z^2 = e^{-8\pi i /5} = e^{2\pi i /5} $. Therefore, the expression becomes $ e^{4\pi i /5} + e^{-4\pi i /5} + e^{-2\pi i /5} + e^{2\pi i /5} $, which is the same as before, so again $ 2\cos(4\pi/5) + 2\cos(2\pi/5) $, which is again -1. Therefore, regardless of which root we take, the value is -1.

Therefore, I think the answer is definitely -1. Therefore, the answer is \boxed{-1}.

**Final Answer**
\boxed{-1}

We are given that $ z $ is a complex number such that $ z^5 = 1 $ and $ z \neq 1 $. We are asked to compute the expression:

$$
z + \frac{1}{z} + z^2 + \frac{1}{z^2}
$$

---

### Step 1: Recognize the nature of $ z $

Since $ z^5 = 1 $, $ z $ is a **5th root of unity**. The 5th roots of unity are the complex numbers $ z = e^{2\pi i k/5} $ for $ k = 0, 1, 2, 3, 4 $. However, since $ z \neq 1 $, we are considering the **non-trivial** 5th roots of unity, i.e., $ z = e^{2\pi i k/5} $ for $ k = 1, 2, 3, 4 $.

---

### Step 2: Use the identity $ \frac{1}{z} = z^4 $

Because $ z^5 = 1 $, we can write:

$$
\frac{1}{z} = z^4 \quad \text{and} \quad \frac{1}{z^2} = z^3
$$

So the original expression becomes:

$$
z + \frac{1}{z} + z^2 + \frac{1}{z^2} = z + z^4 + z^2 + z^3
$$

---

### Step 3: Use the sum of all 5th roots of unity

The sum of all 5th roots of unity is:

$$
1 + z + z^2 + z^3 + z^4 = 0
$$

Subtracting 1 from both sides gives:

$$
z + z^2 + z^3 + z^4 = -1
$$

Thus, the expression simplifies to:

$$
z + z^2 + z^3 + z^4 = -1
$$

---

### Final Answer

$$
\boxed{-1}
$$

[Assistant]
Explanation:
Okay, so I need to compute the value of $ z + \frac{1}{z} + z^2 + \frac{1}{z^2} $ where $ z $ is a complex number satisfying $ z^5 = 1 $ and $ z \neq 1 $. Let me think about how to approach this.

First, since $ z^5 = 1 $, $ z $ is a 5th root of unity. The 5th roots of unity are the solutions to the equation $ z^5 - 1 = 0 $. These can be expressed in the form $ e^{2\pi i k/5} $ for $ k = 0, 1, 2, 3, 4 $. But since $ z \neq 1 $, we know that $ z $ is one of the primitive 5th roots of unity, meaning $ k = 1, 2, 3, 4 $.

Now, the problem is asking for $ z + \frac{1}{z} + z^2 + \frac{1}{z^2} $. Let me note that $ \frac{1}{z} $ is the complex conjugate of $ z $ if $ z $ lies on the unit circle, which it does because all roots of unity lie on the unit circle. Wait, but since $ z^5 = 1 $, then $ |z| = 1 $, so indeed $ \frac{1}{z} = \overline{z} $. However, I don't know if that's immediately helpful here. Maybe I can express $ \frac{1}{z} $ as $ z^4 $, since $ z^5 = 1 $ implies $ z^5 - 1 = 0 $, so $ z^5 = 1 $, so $ z^4 = \frac{1}{z} $. Similarly, $ \frac{1}{z^2} = z^{3} $. Let me check that:

Since $ z^5 = 1 $, multiplying both sides by $ z^{-2} $ gives $ z^{3} = z^{-2} $, so $ \frac{1}{z^2} = z^{3} $. Similarly, $ \frac{1}{z} = z^4 $. Therefore, the original expression can be rewritten as:

$ z + z^4 + z^2 + z^3 $.

So the problem reduces to computing $ z + z^2 + z^3 + z^4 $.

But wait, since $ z^5 = 1 $, the sum of all 5th roots of unity is 0. That is, $ 1 + z + z^2 + z^3 + z^4 = 0 $. Therefore, $ z + z^2 + z^3 + z^4 = -1 $. Therefore, the original expression is equal to -1? Wait, but let me check again.

Wait, the original expression was $ z + \frac{1}{z} + z^2 + \frac{1}{z^2} $, which we converted to $ z + z^4 + z^2 + z^3 $, which is indeed $ z + z^2 + z^3 + z^4 $, and since the sum of all roots is 0, that sum is -1. Therefore, the answer is -1? But let me verify this with an example.

Let me take a specific 5th root of unity and compute the value. Let's take $ z = e^{2\pi i /5} $. Then compute $ z + 1/z + z^2 + 1/z^2 $.

First, $ 1/z = e^{-2\pi i /5} $, and $ 1/z^2 = e^{-4\pi i /5} $. So the expression becomes:

$ e^{2\pi i /5} + e^{-2\pi i /5} + e^{4\pi i /5} + e^{-4\pi i /5} $.

These are the real parts of the complex numbers. Let me compute each term:

$ e^{2\pi i /5} + e^{-2\pi i /5} = 2\cos(2\pi/5) $

Similarly, $ e^{4\pi i /5} + e^{-4\pi i /5} = 2\cos(4\pi/5) $

Therefore, the total expression is $ 2\cos(2\pi/5) + 2\cos(4\pi/5) $.

Now, let me compute this numerically. Let me recall that $ \cos(4\pi/5) = \cos(\pi - \pi/5) = -\cos(\pi/5) $, and $ \cos(2\pi/5) $ is just $ \cos(2\pi/5) $. Therefore, the expression becomes:

$ 2\cos(2\pi/5) - 2\cos(\pi/5) $.

Hmm, so I need to compute this value. Let me recall exact values for these cosines. Alternatively, maybe I can relate them to known quantities.

Alternatively, perhaps I can use the identity that the sum $ z + z^2 + z^3 + z^4 = -1 $, so according to that, the answer is -1. But according to the numerical calculation, it's $ 2\cos(2\pi/5) + 2\cos(4\pi/5) $. Let me compute this numerically.

First, compute $ \cos(2\pi/5) $. Let me recall that $ 2\pi/5 $ is 72 degrees. The cosine of 72 degrees is approximately 0.3090. Similarly, $ \cos(4\pi/5) $ is $ \cos(144^\circ) $, which is approximately -0.8090. Therefore, substituting:

$ 2*(0.3090) + 2*(-0.8090) = 0.618 - 1.618 = -1.0 $. Oh! So that gives -1. So that matches the previous conclusion. Therefore, the answer is indeed -1. Therefore, the original expression is equal to -1.

But let me check again. Wait, so according to the sum of roots, since $ 1 + z + z^2 + z^3 + z^4 = 0 $, so $ z + z^2 + z^3 + z^4 = -1 $. Therefore, the expression is -1. Therefore, the answer is \boxed{-1}?

But let me think again. Wait, the problem says "Compute $ z + \frac{1}{z} + z^2 + \frac{1}{z^2} $", which we converted to $ z + z^4 + z^2 + z^3 $, which is indeed the sum of all roots except 1, so that sum is -1. Therefore, the answer is -1. Therefore, the answer is \boxed{-1}.

But let me check again with another approach. Maybe using algebra.

Since $ z^5 = 1 $, so $ z \neq 1 $, so $ z^4 + z^3 + z^2 + z + 1 = 0 $, so $ z^4 + z^3 + z^2 + z = -1 $. Therefore, the expression $ z + z^2 + z^3 + z^4 = -1 $. Therefore, the answer is -1. Therefore, regardless of which root we take, the value is -1. Therefore, the answer is \boxed{-1}.

But let me check if there's another way to compute this expression. For example, maybe by squaring or something else. Let me try.

Let me denote $ S = z + \frac{1}{z} + z^2 + \frac{1}{z^2} $. Let me note that $ z + \frac{1}{z} = z + z^4 $, and $ z^2 + \frac{1}{z^2} = z^2 + z^3 $. Therefore, S is equal to $ (z + z^4) + (z^2 + z^3) $, which is again the same as before. Alternatively, maybe I can compute $ (z + 1/z)^2 $ or something like that.

Let me compute $ (z + 1/z)^2 = z^2 + 2 + 1/z^2 $. Therefore, $ z^2 + 1/z^2 = (z + 1/z)^2 - 2 $. Similarly, if I let $ A = z + 1/z $, then $ A^2 = z^2 + 2 + 1/z^2 $, so $ z^2 + 1/z^2 = A^2 - 2 $. Therefore, the original expression S is $ A + (A^2 - 2) $. Wait, no. Wait, S is $ z + 1/z + z^2 + 1/z^2 $, which is $ A + (z^2 + 1/z^2) $. Which is $ A + (A^2 - 2) $. Therefore, S = A + A^2 - 2. But then I need to find A.

Alternatively, maybe compute A first. Let me compute $ A = z + 1/z = z + z^4 $. Let me compute $ A $. Let me note that $ z^5 = 1 $, so $ z \neq 0 $, so we can write $ A = z + z^4 $. Similarly, since $ z^4 = z^{-1} $, so $ A = z + z^{-1} $. Let me compute $ A $.

Alternatively, since $ z^5 = 1 $, then $ z^5 - 1 = 0 $, which factors as $ (z - 1)(z^4 + z^3 + z^2 + z + 1) = 0 $. Since $ z \neq 1 $, then $ z^4 + z^3 + z^2 + z + 1 = 0 $, so $ z^4 + z^3 + z^2 + z = -1 $, which is the same as before.

Alternatively, maybe compute $ A = z + z^4 $. Let me compute $ A + z^2 + z^3 = -1 $. But that's the same as before.

Alternatively, maybe compute $ A = z + z^4 $. Let me compute $ A $:

Let me note that $ z^4 = \overline{z} $, since $ |z| = 1 $, so $ \overline{z} = 1/z $. Therefore, $ A = z + \overline{z} = 2\Re(z) $. Similarly, $ z^2 + z^3 = z^2 + z^{-2} = 2\Re(z^2) $. Therefore, S = 2\Re(z) + 2\Re(z^2). But maybe this is more complicated.

Alternatively, since we already have that S = -1, maybe that's the simplest way. But let me check if there's another way to compute S.

Alternatively, let me compute $ S = z + z^2 + z^3 + z^4 = -1 $, so that's straightforward. Therefore, the answer is -1. Therefore, I think that's the answer.

But let me check again with another example. Let me take $ z = e^{2\pi i /5} $, compute $ z + 1/z + z^2 + 1/z^2 $.

As before, $ z = e^{2\pi i /5} $, so $ 1/z = e^{-2\pi i /5} $, $ z^2 = e^{4\pi i /5} $, $ 1/z^2 = e^{-4\pi i /5} $. Therefore, adding them up:

$ e^{2\pi i /5} + e^{-2\pi i /5} + e^{4\pi i /5} + e^{-4\pi i /5} $.

Which is $ 2\cos(2\pi/5) + 2\cos(4\pi/5) $. As before, which is $ 2\cos(72^\circ) + 2\cos(144^\circ) $.

Now, let me compute these values numerically:

First, $ \cos(72^\circ) \approx 0.3090 $, so 2*0.3090 ≈ 0.618.

Then, $ \cos(144^\circ) = \cos(180^\circ - 36^\circ) = -\cos(36^\circ) ≈ -0.8090 $, so 2*(-0.8090) ≈ -1.618.

Adding them together: 0.618 - 1.618 = -1.0. So that's exactly -1. Therefore, this numerical example confirms that the value is indeed -1.

Therefore, regardless of the approach, the answer is -1. Therefore, I think the answer is \boxed{-1}.

But just to make sure, let me think if there's any possible mistake here. For example, could there be a case where $ z + 1/z + z^2 + 1/z^2 $ is different? But since all the non-1 roots satisfy the same equation, and the sum is always -1, regardless of which root we take. For example, take $ z = e^{4\pi i /5} $, which is another root. Then $ 1/z = e^{-4\pi i /5} $, and $ z^2 = e^{8\pi i /5} = e^{-2\pi i /5} $, and $ 1/z^2 = e^{-8\pi i /5} = e^{2\pi i /5} $. Therefore, the expression becomes $ e^{4\pi i /5} + e^{-4\pi i /5} + e^{-2\pi i /5} + e^{2\pi i /5} $, which is the same as before, so again $ 2\cos(4\pi/5) + 2\cos(2\pi/5) $, which is again -1. Therefore, regardless of which root we take, the value is -1.

Therefore, I think the answer is definitely -1. Therefore, the answer is \boxed{-1}.

**Final Answer**
\boxed{-1}

We are given that $ z $ is a complex number such that $ z^5 = 1 $ and $ z \neq 1 $. We are asked to compute the expression:

$$
z + \frac{1}{z} + z^2 + \frac{1}{z^2}
$$

---

### Step 1: Recognize the nature of $ z $

Since $ z^5 = 1 $, $ z $ is a **5th root of unity**. The 5th roots of unity are the complex numbers $ z = e^{2\pi i k/5} $ for $ k = 0, 1, 2, 3, 4 $. However, since $ z \neq 1 $, we are considering the **non-trivial** 5th roots of unity, i.e., $ z = e^{2\pi i k/5} $ for $ k = 1, 2, 3, 4 $.

---

### Step 2: Use the identity $ \frac{1}{z} = z^4 $

Because $ z^5 = 1 $, we can write:

$$
\frac{1}{z} = z^4 \quad \text{and} \quad \frac{1}{z^2} = z^3
$$

So the original expression becomes:

$$
z + \frac{1}{z} + z^2 + \frac{1}{z^2} = z + z^4 + z^2 + z^3
$$

---

### Step 3: Use the sum of all 5th roots of unity

The sum of all 5th roots of unity is:

$$
1 + z + z^2 + z^3 + z^4 = 0
$$

Subtracting 1 from both sides gives:

$$
z + z^2 + z^3 + z^4 = -1
$$

Thus, the expression simplifies to:

$$
z + z^2 + z^3 + z^4 = -1
$$

---

### Final Answer

$$
\boxed{-1}
$$

Solution:
Since $z^5 = 1,$ $z^5 - 1 = 0,$ which factors as
\[(z - 1)(z^4 + z^3 + z^2 + z + 1) = 0.\]Since $z \neq 1,$ $z^4 + z^3 + z^2 + z + 1 = 0.$

Then
\[z + \frac{1}{z} + z^2 + \frac{1}{z^2} = \frac{z^3 + z + z^4 + 1}{z^2} = \frac{-z^2}{z^2} = \boxed{-1}.\]