To prove that \(\mathbb{Q}(\sqrt{2}, \sqrt[3]{2}) = \mathbb{Q}(\sqrt[6]{2})\), we need to show that the field extensions are equal, meaning every element of \(\mathbb{Q}(\sqrt{2}, \sqrt[3]{2})\) can be expressed as an element of \(\mathbb{Q}(\sqrt[6]{2})\) and vice versa.

### 1. Show \(\mathbb{Q}(\sqrt{2}, \sqrt[3]{2}) \subseteq \mathbb{Q}(\sqrt[6]{2})\)

Start by expressing \(\sqrt{2}\) and \(\sqrt[3]{2}\) in terms of \(\sqrt[6]{2}\):
- \(\sqrt{2} = (\sqrt[6]{2})^3\)
- \(\sqrt[3]{2} = (\sqrt[6]{2})^2\)

Since \(\sqrt{2}\) and \(\sqrt[3]{2}\) are both expressible as powers of \(\sqrt[6]{2}\), any element in \(\mathbb{Q}(\sqrt{2}, \sqrt[3]{2})\) can be expressed in terms of \(\sqrt[6]{2}\). Thus, \(\mathbb{Q}(\sqrt{2}, \sqrt[3]{2}) \subseteq \mathbb{Q}(\sqrt[6]{2})\).

### 2. Show \(\mathbb{Q}(\sqrt[6]{2}) \subseteq \mathbb{Q}(\sqrt{2}, \sqrt[3]{2})\)

Consider an arbitrary element in \(\mathbb{Q}(\sqrt[6]{2})\). It can be written as a polynomial expression \(f(\sqrt[6]{2})\) with coefficients in \(\mathbb{Q}\). Specifically, since \(\sqrt[6]{2}\) is a root of the minimal polynomial \(x^6 - 2 = 0\), elements in \(\mathbb{Q}(\sqrt[6]{2})\) can be expressed as:
\[
a_0 + a_1 \sqrt[6]{2} + a_2 (\sqrt[6]{2})^2 + \dots + a_5 (\sqrt[6]{2})^5
\]
where \(a_0, a_1, \dots, a_5 \in \mathbb{Q}\).

Since \(\sqrt[6]{2} = \sqrt[3]{2} \cdot \sqrt{2}\), and both \(\sqrt{2}\) and \(\sqrt[3]{2}\) are in \(\mathbb{Q}(\sqrt{2}, \sqrt[3]{2})\), it follows that \(\sqrt[6]{2}\) is also in \(\mathbb{Q}(\sqrt{2}, \sqrt[3]{2})\). Therefore, every element in \(\mathbb{Q}(\sqrt[6]{2})\) can be expressed as an element in \(\mathbb{Q}(\sqrt{2}, \sqrt[3]{2})\).

Thus, \(\mathbb{Q}(\sqrt[6]{2}) \subseteq \mathbb{Q}(\sqrt{2}, \sqrt[3]{2})\).

### Conclusion
Since \(\mathbb{Q}(\sqrt{2}, \sqrt[3]{2}) \subseteq \mathbb{Q}(\sqrt[6]{2})\) and \(\mathbb{Q}(\sqrt[6]{2}) \subseteq \mathbb{Q}(\sqrt{2}, \sqrt[3]{2})\), we conclude that:
\[
\mathbb{Q}(\sqrt{2}, \sqrt[3]{2}) = \mathbb{Q}(\sqrt[6]{2})
\]

Do you want details or have any questions?

Here are some related questions:
1. What is a field extension, and how is it related to the concept of a basis?
2. How can the minimal polynomial of an algebraic element be determined?
3. What are some examples of other field extensions similar to \(\mathbb{Q}(\sqrt{2}, \sqrt[3]{2})\)?
4. How does Galois theory apply to field extensions like \(\mathbb{Q}(\sqrt{2}, \sqrt[3]{2})\)?
5. What are the degrees of the field extensions \(\mathbb{Q}(\sqrt[6]{2})/\mathbb{Q}\) and \(\mathbb{Q}(\sqrt{2}, \sqrt[3]{2})/\mathbb{Q}\)?

**Tip:** Understanding the structure of field extensions can significantly aid in recognizing and proving equalities between different algebraic structures.

Learn how to prove that the field extensions Q(sqrt(2), cuberoot(2)) and Q(sixthroot(2)) are equal using algebraic techniques. Understand the concepts of field extensions, minimal polynomials, and their applications in algebra. Suitable for advanced high school students and beyond.

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