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The problem provided asks to simplify the expression \( \frac{1}{a^{5/8}} \).

To simplify:

\[
\frac{1}{a^{5/8}} = a^{-5/8}
\]

Now, we need to match this with one of the provided answer choices. Let's inspect them:

1. \( \frac{\sqrt[8]{a^5}}{a} \)
2. \( \frac{\sqrt[8]{a^3}}{a} \)
3. \( \frac{a^{3/8}}{a} \)

Let's analyze each option:

1. \( \frac{\sqrt[8]{a^5}}{a} = \frac{a^{5/8}}{a} = a^{5/8 - 1} = a^{-3/8} \)
2. \( \frac{\sqrt[8]{a^3}}{a} = \frac{a^{3/8}}{a} = a^{3/8 - 1} = a^{-5/8} \)
3. \( \frac{a^{3/8}}{a} = a^{3/8 - 1} = a^{-5/8} \)

Thus, the correct answer is **Option 2**: \( \frac{\sqrt[8]{a^3}}{a} \).

Would you like further details or have any other questions? Here are five related questions:

1. How do you simplify expressions with fractional exponents?
2. What is the general rule for dividing exponents with the same base?
3. Can you explain the difference between \( a^{m/n} \) and \( \sqrt[n]{a^m} \)?
4. How do you convert negative exponents into positive ones?
5. What is the simplified form of \( \frac{a^{3/8}}{a^{1/4}} \)?

**Tip:** When simplifying expressions involving exponents, always apply the exponent rules carefully, especially with fractional and negative exponents!

If 1 / a^(5/8) ∈ ℝ, then its simplified form is equal to one of the given options.

Learn how to simplify the expression 1 / a^(5/8) with step-by-step explanation. This problem involves exponentiation rules and fractional exponents, suitable for students in Grades 9-12.

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