The problem asks for an equivalent expression to $x^{2/3} + \frac{1}{x^{2/3}}$. Let’s solve this step by step.

We are asked to simplify:

$x^{2/3} + \frac{1}{x^{2/3}}$

Step 1: Combine the terms

Notice that this is a sum of two terms with the same base $x^{2/3}$. You can't directly combine the two terms, but the expression can be analyzed in other forms.

Step 2: Look for matching options

We will analyze the answer choices given in the image:

Option A:

$\frac{x^{2/3} + 1}{x^{2/3}}$ This simplifies to: $1 + \frac{1}{x^{2/3}}$ which does not match the original expression.

Option B:

$\frac{x^{4/9} + 1}{x^{2/3}}$ This introduces a term $x^{4/9}$, which does not appear in the original expression. Thus, this is not equivalent.

Option C:

1, which does not match the original expression.

Option D:

$\frac{x + 1}{x^{2/3}}$ This expression introduces $x$ in the numerator, which is not part of the original expression.

Option E:

$\frac{x^{4/3} + 1}{x^{2/3}}$ This simplifies to: $x^{4/3 - 2/3} + \frac{1}{x^{2/3}} = x^{2/3} + \frac{1}{x^{2/3}}$ This is exactly the same as the original expression!

Conclusion:

The correct answer is Option E: $\frac{x^{4/3} + 1}{x^{2/3}}$.

Would you like more details on any of the steps?

Here are 5 related questions for further practice:

1. Simplify $\frac{x^{5/3} + 1}{x^{2/3}}$.
2. What is the result of $\frac{x^{7/3} + 1}{x^{2/3}}$?
3. Simplify $x^{3/2} + \frac{1}{x^{3/2}}$.
4. Find an equivalent expression to $x^{a} + \frac{1}{x^a}$.
5. What is the derivative of $x^{2/3} + \frac{1}{x^{2/3}}$?

Tip: When simplifying expressions with exponents, always check if terms can be factored or simplified using exponent rules.