how do you solve y= the fourth root of x^3-1

To solve the equation \( y = \sqrt[4]{x^3 - 1} \), we need to isolate \( x \) in terms of \( y \). Here's the step-by-step process:

1. **Rewrite the equation**:
   \[
   y = \sqrt[4]{x^3 - 1}
   \]
   This is equivalent to:
   \[
   y = (x^3 - 1)^{1/4}
   \]

2. **Raise both sides to the power of 4** to eliminate the fourth root:
   \[
   y^4 = x^3 - 1
   \]

3. **Solve for \( x^3 \)** by adding 1 to both sides:
   \[
   x^3 = y^4 + 1
   \]

4. **Take the cube root** of both sides to solve for \( x \):
   \[
   x = \sqrt[3]{y^4 + 1}
   \]

Thus, the solution for \( x \) in terms of \( y \) is:
\[
x = \sqrt[3]{y^4 + 1}
\]

### Do you need any further clarification or details?

Here are five related questions to expand your understanding:

1. How would you solve the inverse equation \( y = \sqrt[3]{x^4 + 1} \)?
2. What are the domain and range of the function \( y = \sqrt[4]{x^3 - 1} \)?
3. How would the solution change if the original equation involved a square root instead of a fourth root?
4. Can you differentiate the function \( y = \sqrt[4]{x^3 - 1} \) with respect to \( x \)?
5. How would you graph the function \( y = \sqrt[4]{x^3 - 1} \)?

### Tip:
When working with roots, always check whether the expression inside the root is valid for the range of the function (i.e., non-negative for even roots).

how do you solve y = the fourth root of x^3 - 1

This page explains how to solve the equation y = the fourth root of x^3 - 1, providing a step-by-step process to isolate x in terms of y. The solution involves algebraic manipulation of radicals and roots, including inverse functions. Suitable for high school students in Grades 10-12.

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