Final answer:
The student's question concerns the laws of exponents, specifically which expressions are equivalent to \(\frac{1}{4^3}\). The correct equivalent expression using exponent laws is \(4^{-3}\), which can also be written as \(4^{-2} \cdot 4^{-1}\) because when multiplied, the exponents are added.
Step-by-step explanation:
The question seems to be asking about the laws of exponents and which expressions are equivalent to \(\frac{1}{4^3}\). Understanding exponent laws is crucial for simplifying and manipulating expressions involving powers. The expression \(\frac{1}{4^3}\) simplifies to \(4^{-3}\) by using the division of exponentials. Specifically, when we divide, we subtract the exponents, so if we had \(a^m \div a^n\), it would be \(a^{m-n}\). Now, if we were to evaluate some examples such as \(10^6 \div 10^3\), it would result in \(10^{6-3} = 10^3\).
In the case provided, the expression \(4^{-2} \cdot 4^{-1}\) uses the product of exponents rule. When multiplying exponential terms with the same base, we should add the exponents: \(4^{-2} \cdot 4^{-1} = 4^{-2 + (-1)} = 4^{-3}\), which is indeed equivalent to \(\frac{1}{4^3}\).