To solve the expression [tex]\(\left(\frac{216}{2197}\right)^{-\frac{2}{3}}\)[/tex], follow these steps:
1. Understand the negative exponent:
The negative exponent [tex]\(-\frac{2}{3}\)[/tex] indicates that we need to take the reciprocal of the base and then raise it to the power of [tex]\(\frac{2}{3}\)[/tex].
[tex]\[
\left(\frac{a}{b}\right)^{-\frac{2}{3}} = \left(\frac{b}{a}\right)^{\frac{2}{3}}
\][/tex]
Therefore, [tex]\(\left(\frac{216}{2197}\right)^{-\frac{2}{3}}\)[/tex] can be rewritten as [tex]\(\left(\frac{2197}{216}\right)^{\frac{2}{3}}\)[/tex].
2. Cube root and square the result:
The exponent of [tex]\(\frac{2}{3}\)[/tex] can be split into two parts: the cube root and squaring. Thus, we first take the cube root of [tex]\(\frac{2197}{216}\)[/tex], and then square the result.
Let's find the cube root first:
- The cube root of 2197 is 13, because [tex]\(13^3 = 2197\)[/tex].
- The cube root of 216 is 6, because [tex]\(6^3 = 216\)[/tex].
Therefore,
[tex]\[
\sqrt[3]{\frac{2197}{216}} = \frac{13}{6}
\][/tex]
3. Square the result:
Now, we need to square [tex]\(\frac{13}{6}\)[/tex]:
[tex]\[
\left(\frac{13}{6}\right)^2 = \frac{13^2}{6^2} = \frac{169}{36}
\][/tex]
Thus, the expression [tex]\(\left(\frac{216}{2197}\right)^{-\frac{2}{3}}\)[/tex] evaluates to [tex]\(\frac{169}{36}\)[/tex]. Since this fraction is in its simplest form, no further simplification is needed.
Hence, the final simplified fractional answer is:
[tex]\[
\boxed{\frac{169}{36}}
\][/tex]
