To simplify the expression

\[ \frac{\sqrt{2}}{2 \sqrt[4]{2}} \]

we will follow these steps:

1. Simplify the radicals if possible.
2. Combine the radicals in a single form.
3. Rationalize the denominator if necessary.

### Step-by-Step Solution

1. **Simplify the radicals:**

Given:
\[ \frac{\sqrt{2}}{2 \sqrt[4]{2}} \]

2. **Express radicals with the same index:**

We know that:
\[ \sqrt{2} = 2^{1/2} \]
\[ \sqrt[4]{2} = 2^{1/4} \]

So, the expression becomes:
\[ \frac{2^{1/2}}{2 \cdot 2^{1/4}} \]

3. **Combine the exponents in the denominator:**

\[ 2 \cdot 2^{1/4} = 2^{1 + 1/4} = 2^{5/4} \]

Now the expression is:
\[ \frac{2^{1/2}}{2^{5/4}} \]

4. **Simplify the fraction:**

\[ \frac{2^{1/2}}{2^{5/4}} = 2^{1/2 - 5/4} \]

Convert the exponents to have a common denominator:
\[ \frac{1}{2} = \frac{2}{4} \]
\[ 2^{1/2 - 5/4} = 2^{2/4 - 5/4} = 2^{-3/4} \]

5. **Express the result with a positive exponent:**

\[ 2^{-3/4} = \frac{1}{2^{3/4}} \]

6. **Write in radical form:**

\[ 2^{3/4} = (\sqrt[4]{2})^3 \]

So the expression becomes:
\[ \frac{1}{(\sqrt[4]{2})^3} = \frac{1}{\sqrt[4]{8}} \]

Thus, the simplified expression is:

\[ \frac{1}{\sqrt[4]{8}} \]

### Final Answer:

\[ \frac{1}{\sqrt[4]{8}} \]

Do you have any questions or need further details on any of the steps?

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Here are 8 related questions:

1. How do you simplify an expression with mixed radical indices?
2. What is the process for converting radicals to exponential form?
3. How do you simplify the expression \(2^{1/2 - 5/4}\)?
4. What is the significance of rationalizing exponents in expressions?
5. How do you handle negative exponents in simplification?
6. What are the steps to combine radicals under a single form?
7. How do you express \(2^{3/4}\) in radical notation?
8. What is the value of \(\sqrt[4]{8}\)?

**Tip:** When dealing with different indices in radicals, it often helps to convert them to exponential form to simplify the arithmetic operations.

Learn how to simplify the expression √2/(2∜2) step-by-step with our detailed solution. Understand radicals, exponents, and simplification techniques. Suitable for high school students.

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