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The given function is \( y = (5x^2 - 1)^{\frac{4}{3}} \).

This is a composite function, where you have a function of \( x \) raised to a fractional power. Since you are asked to "solve" it, I assume you are looking for either its simplification, derivative, or a particular operation to perform on it. Here are possible interpretations:

### 1. **Simplification**:
   The function is already in its simplest form, as the expression \( (5x^2 - 1)^{\frac{4}{3}} \) cannot be simplified further without specific values of \( x \).

### 2. **Finding the Derivative**:
   If the goal is to differentiate \( y \) with respect to \( x \), you would use the chain rule. The steps are as follows:
   
   - Let \( u = 5x^2 - 1 \), so \( y = u^{\frac{4}{3}} \).
   - The derivative of \( u^{\frac{4}{3}} \) with respect to \( u \) is:
     \[
     \frac{d}{du} \left( u^{\frac{4}{3}} \right) = \frac{4}{3} u^{\frac{1}{3}}
     \]
   - Now, differentiate \( u = 5x^2 - 1 \) with respect to \( x \):
     \[
     \frac{du}{dx} = 10x
     \]
   - Finally, applying the chain rule:
     \[
     \frac{dy}{dx} = \frac{4}{3} (5x^2 - 1)^{\frac{1}{3}} \cdot 10x
     \]
     Simplifying:
     \[
     \frac{dy}{dx} = \frac{40x}{3} (5x^2 - 1)^{\frac{1}{3}}
     \]

### 3. **Finding the Inverse**:
   If you're looking for the inverse function, solving for \( x \) in terms of \( y \), you would proceed as follows:
   
   \[
   y = (5x^2 - 1)^{\frac{4}{3}}
   \]
   - Raise both sides to the power of \( \frac{3}{4} \):
     \[
     y^{\frac{3}{4}} = 5x^2 - 1
     \]
   - Solve for \( x^2 \):
     \[
     x^2 = \frac{y^{\frac{3}{4}} + 1}{5}
     \]
   - Take the square root:
     \[
     x = \pm \sqrt{\frac{y^{\frac{3}{4}} + 1}{5}}
     \]

Would you like a deeper explanation for any specific approach?

### Related Questions:
1. How do you apply the chain rule to more complex functions?
2. How do you solve composite functions by finding their inverses?
3. What are the typical steps to differentiate functions involving fractional exponents?
4. Can this function be integrated easily? What method would you use?
5. How would you handle this function if you were asked to evaluate it for specific values of \( x \)?

### Tip:
When differentiating composite functions like this, always apply the chain rule carefully by differentiating the outer function first and then multiplying by the derivative of the inner function.

Solve the composite function y = (5x^2 - 1)^(4/3)

Understand how to solve the composite function y = (5x^2 - 1)^(4/3) using differentiation. This solution applies the chain rule and power rule to derive the function's derivative, covering key algebraic and calculus concepts for advanced high school students.

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