To simplify $[3^{(5/3)}]^{-6}$, follow these steps:

Step 1: Apply the Power Rule

$\left(a^m\right)^n = a^{m \cdot n}$ Here, $a = 3$, $m = \frac{5}{3}$, and $n = -6$.

$[3^{(5/3)}]^{-6} = 3^{\frac{5}{3} \cdot (-6)} = 3^{-10}$

Step 2: Simplify the Exponent

$\frac{5}{3} \cdot (-6) = -10$ Thus, $3^{\frac{5}{3} \cdot (-6)} = 3^{-10}.$

Step 3: Write the Final Simplified Form

The simplified expression is: $3^{-10} = \frac{1}{3^{10}}$

So, the final result is: $\boxed{\frac{1}{3^{10}}}$

Would you like further explanations or examples of similar problems?

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Related Questions:

1. What is $3^{-10}$ expressed as a decimal approximation?
2. How do you simplify $[x^{(7/4)}]^3$?
3. How is the power rule $(a^m)^n = a^{m \cdot n}$ derived?
4. Simplify $2^{(4/5) \cdot 10}$.
5. What are the properties of exponents involving fractional bases?

Tip:

When dealing with negative exponents, remember that $a^{-n} = \frac{1}{a^n}$.