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We are asked to find an expression equivalent to the given one, which is:

\[
\frac{1}{(3x^2 + 5)^{\frac{5}{2}}}
\]

### Step-by-Step Simplification:

1. **Analyze the Original Expression:**
   \[
   \frac{1}{(3x^2 + 5)^{\frac{5}{2}}}
   \]
   This is the same as having \((3x^2 + 5)^{-\frac{5}{2}}\), by applying the property of exponents \( a^{-b} = \frac{1}{a^b} \).

2. **Rewriting in Radical Form:**
   The exponent \(\frac{5}{2}\) can be split as follows:
   \[
   (3x^2 + 5)^{\frac{5}{2}} = \left((3x^2 + 5)^5\right)^{\frac{1}{2}} = \sqrt{(3x^2 + 5)^5}
   \]
   So, we can rewrite the original expression as:
   \[
   \frac{1}{\sqrt{(3x^2 + 5)^5}}
   \]

3. **Identifying the Correct Answer:**
   Now, looking at the given choices, the option that matches this is:
   \[
   \frac{1}{\sqrt{(3x^2 + 5)^5}}
   \]
   This corresponds to the first choice on the left side of the options.

### Final Answer:
\[
\boxed{\frac{1}{\sqrt{(3x^2 + 5)^5}}}
\]

Would you like further details or clarifications?

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Here are 5 related questions to deepen understanding:

1. How would you simplify \((a + b)^{\frac{3}{2}}\)?
2. What are the properties of exponents that allow fractional exponents to be converted to radicals?
3. How would the expression change if the exponent were \(\frac{7}{3}\)?
4. Can you find an equivalent expression for \((x^2 + 4)^{\frac{3}{4}}\)?
5. How do you simplify a general expression of the form \( \frac{1}{(a+b)^n} \)?

**Tip:** When dealing with fractional exponents, always remember that the numerator represents the power, and the denominator represents the root.

Select the expression that is equivalent to \(\frac{1}{(3x^2 + 5)^{\frac{5}{2}}}\)

Learn how to simplify the expression \(\frac{1}{(3x^2 + 5)^{\frac{5}{2}}}\) using exponent and radical rules. This problem involves transforming fractional exponents into radical form and applying exponent properties. Suitable for Grades 9-11 students.

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