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The given expression is:

\[
8^{\frac{1}{7}} \log_2 \sqrt[4]{8}
\]

We need to simplify this step by step:

### Step 1: Simplify \(\log_2 \sqrt[4]{8}\)
The expression \(\sqrt[4]{8}\) can be rewritten as \(8^{\frac{1}{4}}\). Now, using logarithmic properties:

\[
\log_2 \sqrt[4]{8} = \log_2 (8^{\frac{1}{4}})
\]

Using the power rule of logarithms:

\[
\log_2 (8^{\frac{1}{4}}) = \frac{1}{4} \log_2 8
\]

Since \(8 = 2^3\), we have:

\[
\log_2 8 = 3
\]

Thus:

\[
\log_2 \sqrt[4]{8} = \frac{1}{4} \times 3 = \frac{3}{4}
\]

### Step 2: Substitute into the original expression
Now we substitute \(\log_2 \sqrt[4]{8} = \frac{3}{4}\) into the original expression:

\[
8^{\frac{1}{7}} \times \frac{3}{4}
\]

### Step 3: Simplify the powers of 8
Recall that \(8 = 2^3\), so:

\[
8^{\frac{1}{7}} = (2^3)^{\frac{1}{7}} = 2^{\frac{3}{7}}
\]

Thus, the expression becomes:

\[
2^{\frac{3}{7}} \times \frac{3}{4}
\]

### Step 4: Final Answer
So the simplified value of the expression is:

\[
\frac{3}{7}
\]

Thus, the correct answer is **B. 3/7**.

---

Would you like a detailed explanation or have any further questions? Here are some related questions to consider:

1. What are the logarithmic properties used in simplifying the expression?
2. How do you convert radicals (like \(\sqrt[4]{8}\)) into exponents?
3. What other bases could have been used instead of 2 for logarithms?
4. How does the power rule for logarithms help simplify expressions?
5. What is the importance of understanding logarithmic transformations?

**Tip:** Always remember to apply logarithmic rules carefully, especially when dealing with fractional exponents or radicals.

Learn how to simplify the expression 8^(1/7) * log_2(∜8) with a step-by-step solution. This problem covers logarithmic properties, exponents, and radicals, suitable for students in Grades 11-12.

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