To find the next term of the geometric sequence given by
\[ -\dfrac{128}{27},\ \dfrac{32}{9},\ -\dfrac{8}{3}, \]
we first need to determine the common ratio of the sequence.
The second term divided by the first term gives us the common ratio \( r \):
\[ r = \frac{\frac{32}{9}}{-\frac{128}{27}}. \]
To simplify this, we multiply by the reciprocal of the first term:
\[ r = \frac{32}{9} \cdot \left(-\frac{27}{128}\right). \]
Calculating this gives:
\[ r = -\frac{32 \cdot 27}{9 \cdot 128}. \]
Now simplify \(-\frac{32}{128}\) (which is \(-\frac{1}{4}\)) and \(\frac{27}{9}\) (which is \(3\)):
\[ r = -\frac{1 \cdot 3}{4} = -\frac{3}{4}. \]
Next, we can verify this common ratio by checking the third term divided by the second term:
\[ \frac{-\frac{8}{3}}{\frac{32}{9}} = -\frac{8}{3} \cdot \frac{9}{32} = -\frac{8 \cdot 9}{3 \cdot 32}. \]
Calculating \( -\frac{8}{32}\) (which simplifies to \(-\frac{1}{4}\)) and \(\frac{9}{3}\) (which is \(3\)) gives:
\[ -\frac{3}{4}. \]
The common ratio \( r \) is confirmed as \(-\frac{3}{4}\).
To find the next term \( a_4 \), we multiply the third term by the common ratio:
\[ a_4 = a_3 \cdot r = -\frac{8}{3} \cdot -\frac{3}{4}. \]
Calculating this results in:
\[ a_4 = \frac{8 \cdot 3}{3 \cdot 4} = \frac{8}{4} = 2. \]
Thus, the next term in the geometric sequence is
\[ \boxed{2}. \]
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