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The given expression is:

\[
T = \sqrt{a^{\frac{3}{5}} b^{\frac{4}{5}} c^{\frac{1}{5}}} \cdot \sqrt{b^{\frac{3}{5}} c^{\frac{4}{5}} a^{\frac{1}{5}}} \cdot \sqrt{c^{\frac{3}{5}} a^{\frac{4}{5}} b^{\frac{1}{5}}}
\]
with the condition \(abc = 1\).

### Step 1: Combine the radicals
Since all terms inside the square roots are multiplied, we can combine them under a single square root:

\[
T = \sqrt{ \left(a^{\frac{3}{5}} b^{\frac{4}{5}} c^{\frac{1}{5}}\right) \cdot \left(b^{\frac{3}{5}} c^{\frac{4}{5}} a^{\frac{1}{5}}\right) \cdot \left(c^{\frac{3}{5}} a^{\frac{4}{5}} b^{\frac{1}{5}}\right) }
\]

### Step 2: Multiply the exponents
Now, combine the exponents of each variable. For the exponents of \(a\), \(b\), and \(c\), we add their respective exponents across all terms.

- For \(a\), the exponents are: \( \frac{3}{5}, \frac{1}{5}, \frac{4}{5} \). Adding them gives:
  \[
  \frac{3}{5} + \frac{1}{5} + \frac{4}{5} = \frac{8}{5}
  \]

- For \(b\), the exponents are: \( \frac{4}{5}, \frac{3}{5}, \frac{1}{5} \). Adding them gives:
  \[
  \frac{4}{5} + \frac{3}{5} + \frac{1}{5} = \frac{8}{5}
  \]

- For \(c\), the exponents are: \( \frac{1}{5}, \frac{4}{5}, \frac{3}{5} \). Adding them gives:
  \[
  \frac{1}{5} + \frac{4}{5} + \frac{3}{5} = \frac{8}{5}
  \]

Thus, the expression inside the square root becomes:

\[
T = \sqrt{a^{\frac{8}{5}} b^{\frac{8}{5}} c^{\frac{8}{5}}}
\]

### Step 3: Simplify using \(abc = 1\)
Since we are given that \(abc = 1\), this means \( a^{\frac{1}{5}} b^{\frac{1}{5}} c^{\frac{1}{5}} = 1 \). Therefore, the entire expression simplifies to:

\[
T = \sqrt{(abc)^{\frac{8}{5}}} = \sqrt{1^{\frac{8}{5}}} = \sqrt{1} = 1
\]

Thus, the value of \(T\) is \( \boxed{1} \).

---

Would you like more details or have any questions?

Here are 5 related questions you might find useful:
1. What happens if the value of \( abc \neq 1 \)?
2. How would the result change if the exponents inside the roots were different?
3. What mathematical properties allow us to combine the radicals?
4. How does the condition \( abc = 1 \) simplify calculations in problems like this?
5. Can you apply the same method to similar problems involving different roots?

**Tip:** Always check if variables simplify to a known identity (like \( abc = 1 \)) to make complex expressions easier to handle!

Determina el valor de: T = \sqrt{a^{\frac{3}{5}} b^{\frac{4}{5}} c^{\frac{1}{5}}} \cdot \sqrt{b^{\frac{3}{5}} c^{\frac{4}{5}} a^{\frac{1}{5}}} \cdot \sqrt{c^{\frac{3}{5}} a^{\frac{4}{5}} b^{\frac{1}{5}}} si abc = 1.

This problem focuses on simplifying a complex radical expression with exponents, given the condition abc = 1. It covers key concepts like exponent addition and radical multiplication, suitable for high school students (Grades 10-12).

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