1. The problem asks which two expressions are equivalent to $4^{-3}$. 2. Recall that negative exponents mean the reciprocal: $4^{-3} = \frac{1}{4^3}$. 3. Calculate $4^3$: $$4^3 = 4 \times 4 \times 4 = 64$$ 4. So, $4^{-3} = \frac{1}{64}$. 5. Check each expression: - $4 \times 4 \times 4 = 64$ \(\neq \frac{1}{64}\) - $\frac{1}{4^3} = \frac{1}{64}$ \(\text{matches}\) - $\frac{1}{4} \times 3 = \frac{3}{4}$ \(\neq \frac{1}{64}\) - $\frac{1}{3} \times 3 \times 3 \times 3 = \frac{27}{3} = 9$ \(\neq \frac{1}{64}\) - $\frac{1}{4} \times 4 \times 4 = \frac{16}{4} = 4$ \(\neq \frac{1}{64}\) - $\frac{1}{34} \approx 0.0294$ \(\neq \frac{1}{64} = 0.015625\) 6. Therefore, only $\frac{1}{4^3}$ matches exactly. 7. The question says "Which two expressions". None other is exactly equal. But check if $1 / 4 \times 4 \times 4$ might be misinterpreted: - If interpreted as $\frac{1}{4} \times 4 \times 4 = 4$, not $\frac{1}{64}$. 8. Check for typo: Possibly $1 / 4^3$ and $\frac{1}{4}^3$ (which is $\left(\frac{1}{4}\right)^3$) both equal $\frac{1}{64}$. 9. Among given expressions, only $\frac{1}{4^3}$ (second expression) is correct. 10. None of others equals $4^{-3}$. **Final answer:** Only the expression $\frac{1}{4^3}$ equals $4^{-3}$ exactly.