The image contains a problem from mathematics asking to rationalize the expression involving square roots:

[ \text{Rationalize} \ \text{one of} \ \frac{\sqrt[3]{27}}{\sqrt[3]{7}} \times \dots ]

The problem is followed by three different options. Since the full content of the problem is not entirely clear from the image, I'll work with the information provided.

Let's start by analyzing the expression that appears to be in question.

Step 1: Simplify the Expression

Given the expression involves cube roots, let's rewrite and simplify it.

• $\sqrt[3]{27}$ can be simplified since $27 = 3^3$, so $\sqrt[3]{27} = 3$.
• The expression likely involves simplifying $\sqrt[3]{7}$.

Step 2: Rationalizing the Expression

To rationalize an expression like $\frac{\sqrt[3]{27}}{\sqrt[3]{7}}$, you typically multiply both the numerator and denominator by a value that will eliminate the root in the denominator. Since we are dealing with cube roots:

[ \frac{3}{\sqrt[3]{7}} \times \frac{\sqrt[3]{7^2}}{\sqrt[3]{7^2}} = \frac{3\sqrt[3]{49}}{7} ]

Step 3: Simplify the Result

This gives us the final simplified and rationalized form.

Now, from this understanding, you can choose the correct option based on this explanation.

Would you like more details, or have any questions about this problem? Here are five related questions you might find helpful:

1. How do you rationalize denominators with different roots (e.g., square roots, cube roots)?
2. Can you provide more examples of rationalizing cube roots?
3. What are the general rules for simplifying radicals?
4. How does the process differ between square roots and cube roots?
5. What are some common mistakes to avoid when rationalizing expressions?

Tip: Always simplify the radical expressions as much as possible before rationalizing to make the process easier.