Question

Select all of the expressions that are equivalent to (2/3)^−4
Responses

2×−43×−4
the fraction with numerator 2 times negative 4 and denominator 3 times negative 4

(−23)4
open paren negative 2 thirds close paren to the 4th power

(32)4
3 halves to the 4th power

812
8 over 12

−2×2×2×23×3×3×3
negative the fraction with numerator 2 times 2 times 2 times 2 and denominator 3 times 3 times 3 times 3

−1681
negative 16 over 81

8116
81 over 16

3×3×3×32×2×2×2

Answer 1

To determine which expressions are equivalent to \((\frac{2}{3})^{-4}\), we can manipulate the original expression step by step.

1. **Applying the negative exponent rule**:
\[
(a/b)^{-n} = (b/a)^{n}
\]
So,
\[
(\frac{2}{3})^{-4} = (\frac{3}{2})^{4}
\]

2. **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:

1. **\(2^{-4} \times 3^{-4}\)**:
\[
= \frac{1}{2^4} \cdot \frac{1}{3^4} = \frac{1}{16} \cdot \frac{1}{81} = \frac{1}{1296} \quad \text{(not equivalent)}
\]

2. **The fraction with numerator \(2 \times -4\) and denominator \(3 \times -4\)**:
\[
= \frac{-8}{-12} = \frac{8}{12} = \frac{2}{3} \quad \text{(not equivalent)}
\]

3. **\((-2/3)^{4}\)**:
\[
= \frac{16}{81} \quad \text{(not equivalent)}
\]

4. **\((\frac{3}{2})^{4}\)**:
\[
= \frac{81}{16} \quad \text{(equivalent)}
\]

5. **\(\frac{8}{12}\)**:
\[
= \frac{2}{3} \quad \text{(not equivalent)}
\]

6. **\(-\frac{16}{81}\)**:
\[
\text{(not equivalent)}
\]

7. **\(\frac{81}{16}\)**:
\[
= \frac{81}{16} \quad \text{(equivalent)}
\]

8. **\(3 \times 3 \times 3 \times 3 \times 2 \times 2 \times 2 \times 2\)**:
\[
= \frac{81}{16} \quad \text{(equivalent)}
\]

### 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:
- \((\frac{3}{2})^{4}\)
- \(\frac{81}{16}\)
- \(3 \times 3 \times 3 \times 3 \times 2 \times 2 \times 2 \times 2\)