https://file.aiquickdraw.com/mathbot/file/1725973263200nfv7z0ll.jpg

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.

Which expression is equivalent to x^(2/3) + 1 / x^(2/3)?

Learn how to simplify the expression x^(2/3) + 1 / x^(2/3) by applying the rules of exponents. This problem is perfect for high school students and covers important algebraic concepts. We walk you through each step with a detailed explanation.

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