To solve the expression [tex]\(\left(\frac{1}{216}\right)^{-\frac{2}{3}}\)[/tex], we can proceed step-by-step as follows:
1. Understanding the Expression:
- We have a fraction [tex]\(\left(\frac{1}{216}\right)\)[/tex] raised to a negative exponent [tex]\(-\frac{2}{3}\)[/tex].
2. Simplifying the Negative Exponent:
- Recall that a negative exponent indicates a reciprocal. Specifically, [tex]\(x^{-a} = \frac{1}{x^{a}}\)[/tex].
- Hence, [tex]\(\left(\frac{1}{216}\right)^{-\frac{2}{3}}\)[/tex] can be rewritten as [tex]\(\left(\frac{1}{\left(\frac{1}{216}\right)^{-\frac{2}{3}}}\right)\)[/tex], which simplifies to [tex]\(216^{\frac{2}{3}}\)[/tex].
3. Calculating the Positive Exponent:
- To proceed with calculating [tex]\(216^{\frac{2}{3}}\)[/tex], note that taking a power of [tex]\(\frac{2}{3}\)[/tex] is equivalent to taking the cube root first, and then squaring the result, or vice versa.
- This can be mathematically expressed as [tex]\(x^{\frac{a}{b}} = \left(\sqrt[b]{x}\right)^a\)[/tex].
- Therefore, [tex]\(216^{\frac{2}{3}} = \left( \sqrt[3]{216} \right)^2\)[/tex].
4. Finding the Cube Root of 216:
- We need to determine [tex]\(\sqrt[3]{216}\)[/tex]. The cube root of 216 is 6 because [tex]\(6^3 = 216\)[/tex].
- Thus, [tex]\(\sqrt[3]{216} = 6\)[/tex].
5. Squaring the Result:
- The next step is to square the result from the cube root.
- Squaring 6 gives us [tex]\(6^2 = 36\)[/tex].
Therefore, [tex]\(\left(\frac{1}{216}\right)^{-\frac{2}{3}}\)[/tex] simplifies to 36.
Thus, the final result is:
[tex]\[
\boxed{36}
\][/tex]
