To determine which expressions are equivalent to \((\frac{2}{3})^{-4}\), we can manipulate the original expression step by step.
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Applying the negative exponent rule: \[ (a/b)^{-n} = (b/a)^{n} \] So, \[ (\frac{2}{3})^{-4} = (\frac{3}{2})^{4} \]
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Calculating \((\frac{3}{2})^{4}\): \[ (\frac{3}{2})^{4} = \frac{3^{4}}{2^{4}} = \frac{81}{16} \]
Thus, we need to check which of the provided responses are equal to \(\frac{81}{16}\).
Checking the responses:
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\(2^{-4} \times 3^{-4}\): \[ = \frac{1}{2^4} \cdot \frac{1}{3^4} = \frac{1}{16} \cdot \frac{1}{81} = \frac{1}{1296} \quad \text{(not equivalent)} \]
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The fraction with numerator \(2 \times -4\) and denominator \(3 \times -4\): \[ = \frac{-8}{-12} = \frac{8}{12} = \frac{2}{3} \quad \text{(not equivalent)} \]
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\((-2/3)^{4}\): \[ = \frac{16}{81} \quad \text{(not equivalent)} \]
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\((\frac{3}{2})^{4}\): \[ = \frac{81}{16} \quad \text{(equivalent)} \]
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\(\frac{8}{12}\): \[ = \frac{2}{3} \quad \text{(not equivalent)} \]
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\(-\frac{16}{81}\): \[ \text{(not equivalent)} \]
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\(\frac{81}{16}\): \[ = \frac{81}{16} \quad \text{(equivalent)} \]
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\(3 \times 3 \times 3 \times 3 \times 2 \times 2 \times 2 \times 2\): \[ = \frac{81}{16} \quad \text{(equivalent)} \]
Summary of Equivalent Expressions:
- \((\frac{3}{2})^{4}\)
- \(\frac{81}{16}\)
- \(3 \times 3 \times 3 \times 3 \times 2 \times 2 \times 2 \times 2\)
Therefore, the correct equivalents from the options provided are:
- \((\frac{3}{2})^{4}\)
- \(\frac{81}{16}\)
- \(3 \times 3 \times 3 \times 3 \times 2 \times 2 \times 2 \times 2\)