https://file.aiquickdraw.com/mathbot/file/1726691778251zc3ucdbw.jpg

The given equation in the image is:

\[
8x^2 + 8y^5 = 64
\]

We can simplify this by dividing both sides by 8:

\[
x^2 + y^5 = 8
\]

### Symmetry of the Curve:

To check the symmetry of a curve, we generally examine it with respect to:
1. The x-axis
2. The y-axis
3. The origin

- **Symmetry with respect to the x-axis**: Replace \(y\) with \(-y\) in the equation and see if the equation remains unchanged.
  - Original equation: \(x^2 + y^5 = 8\)
  - Replacing \(y\) with \(-y\): \(x^2 + (-y)^5 = x^2 - y^5 = 8\), which is not the same equation. Thus, the curve is **not symmetric about the x-axis**.

- **Symmetry with respect to the y-axis**: Replace \(x\) with \(-x\) in the equation and see if the equation remains unchanged.
  - Original equation: \(x^2 + y^5 = 8\)
  - Replacing \(x\) with \(-x\): \((-x)^2 + y^5 = x^2 + y^5 = 8\), which is the same equation. Thus, the curve is **symmetric about the y-axis**.

- **Symmetry with respect to the origin**: Replace both \(x\) with \(-x\) and \(y\) with \(-y\), and check if the equation remains the same.
  - Replacing \(x\) with \(-x\) and \(y\) with \(-y\): \((-x)^2 + (-y)^5 = x^2 - y^5 = 8\), which is not the same equation. Thus, the curve is **not symmetric about the origin**.

### Conclusion:
The curve \(x^2 + y^5 = 8\) is **symmetric with respect to the y-axis**.

Would you like further details or explanations?

Here are some related questions:
1. What is the general approach to finding the symmetry of curves?
2. How does the degree of a term (like \(y^5\)) affect symmetry?
3. Can this equation be classified as a common type of curve, like an ellipse or parabola?
4. How can we graph this equation to better visualize the symmetry?
5. What role does the exponent of \(x^2\) and \(y^5\) play in defining the shape of the curve?

**Tip**: Always check symmetry by systematically replacing variables and verifying if the equation remains unchanged.

The curve 8x^2 + 8y^5 = 64 is symmetric with respect to:

Explore the symmetry of the curve 8x^2 + 8y^5 = 64. This detailed explanation covers how to check symmetry with respect to the x-axis, y-axis, and origin using algebraic techniques. Perfect for high school students (Grades 10-12).

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