To find the \(x\)- and \(y\)-intercepts of the equation \(x^4 - y^3 = 1\), we proceed as follows:

### 1. \(x\)-Intercept:
The \(x\)-intercept occurs where \(y = 0\). To find the \(x\)-intercept, substitute \(y = 0\) into the equation:

\[
x^4 - (0)^3 = 1 \implies x^4 = 1
\]

Solving for \(x\):

\[
x = \pm 1
\]

Thus, the \(x\)-intercepts are:
\[
(1, 0) \quad \text{and} \quad (-1, 0)
\]

### 2. \(y\)-Intercept:
The \(y\)-intercept occurs where \(x = 0\). To find the \(y\)-intercept, substitute \(x = 0\) into the equation:

\[
(0)^4 - y^3 = 1 \implies -y^3 = 1 \implies y^3 = -1
\]

Solving for \(y\):

\[
y = -1
\]

Thus, the \(y\)-intercept is:
\[
(0, -1)
\]

### Summary:
- \(x\)-intercepts: \((1, 0)\) and \((-1, 0)\)
- \(y\)-intercept: \((0, -1)\)

Would you like further details on the graph or any of the steps?

### Related Questions:
1. What is the general shape of the graph for \(x^4 - y^3 = 1\)?
2. How can you find the slope of the tangent at the intercept points?
3. How do you solve \(x^4 = k\) for other values of \(k\)?
4. Can you find more intercepts if you shift the equation?
5. How does the equation change if you add more terms to \(x^4 - y^3 = 1\)?

### Tip:
For complex equations, always check whether intercepts occur by setting each variable to zero first, as it's a straightforward way to simplify.

Discover how to find the x- and y-intercepts of the equation x^4 - y^3 = 1 with a step-by-step explanation. This solution covers algebraic techniques for polynomial equations, including solving for intercepts. Ideal for high school students in Grades 9-12.

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