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What is the denominator of this fraction?

To find the denominator of the given fraction, we need to simplify the expression in the denominator. The fraction is:

\[ \frac{4}{\sqrt[3]{54} - \sqrt[3]{2}} \]

We need to simplify the expression \( \sqrt[3]{54} - \sqrt[3]{2} \).

First, express 54 and 2 as their prime factorizations:

\[ 54 = 2 \times 3^3 \]

Now, we have:

\[ \sqrt[3]{54} = \sqrt[3]{2 \times 3^3} = \sqrt[3]{2} \times \sqrt[3]{3^3} = \sqrt[3]{2} \times 3 \]

So, the denominator becomes:

\[ \sqrt[3]{54} - \sqrt[3]{2} = 3\sqrt[3]{2} - \sqrt[3]{2} \]

Factor out the common term \( \sqrt[3]{2} \):

\[ = \sqrt[3]{2} (3 - 1) = \sqrt[3]{2} \times 2 \]

Therefore, the denominator simplifies to:

\[ 2\sqrt[3]{2} \]

So, the denominator of the given fraction is:

\[ \boxed{2\sqrt[3]{2}} \]

Would you like any details or have any questions?

Here are 8 related questions:
1. How do you find the cube root of a number?
2. What is the process for simplifying radical expressions?
3. How do you factor expressions with cube roots?
4. Can the difference of cube roots be simplified using other algebraic identities?
5. How do you handle cube roots in fractions?
6. What are some common properties of cube roots?
7. How do you rationalize the denominator when it contains cube roots?
8. How can cube roots be applied in real-world problems?

**Tip:** When simplifying expressions with roots, always look for common factors and prime factorizations to make the process easier.

Learn how to find the denominator of the fraction 4 divided by the cube root of 54 minus the cube root of 2. This detailed solution covers simplifying radical expressions and prime factorization techniques. Suitable for advanced high school students.

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