What is the product?
$4^3 \cdot 4^{-3}$
We are evaluating the expression \(4^3 \cdot 4^{-3}\). This problem involves understanding the rules of exponents. The two primary laws of exponents we will utilize are the product rule for exponents and the rule for negative exponents. The product rule states that for any non-zero base \(a\) and exponents \(m\) and \(n\), the product is \(a^m \cdot a^n = a^{m+n}\). The rule for negative exponents states that \(a^{-n} = \frac{1}{a^n}\), which implies \(a^n \cdot a^{-n} = a^n \cdot \frac{1}{a^n} = \frac{a^n}{a^n} = 1\). Additionally, we know that any non-zero base raised to the power of zero is \(1\), i.e., \(a^0 = 1\).
Using the product rule for exponents, we can simplify the expression:
\[4^3 \cdot 4^{-3} = 4^{3 + (-3)}\]
Next, we perform the addition in the exponent:
\[3 + (-3) = 0\]
Thus, the expression becomes:
\[4^0\]
Applying the property that any non-zero base raised to the power of \(0\) equals \(1\), we find:
\[4^0 = 1\]
Alternatively, using the definition of negative exponents:
\[4^3 \cdot 4^{-3} = 4^3 \cdot \frac{1}{4^3} = \frac{4^3}{4^3} = 1\]
Both methods confirm the same result.
We have evaluated the expression \(4^3 \cdot 4^{-3}\) and determined that it is equal to \(1\). Comparing this result with the provided options, we see that the correct choice is \(1\).
The final answer is \(\boxed{1}\).
Exponent rules are fundamental in scientific notation, which is used to express very large or very small numbers compactly. For instance, in physics, the mass of an electron is approximately \(9.11 \times 10^{-31}\) kg. If you were to multiply or divide values with scientific notation, you would rely on these exact exponent properties to simplify the calculation, just as we simplified \(4^3 \cdot 4^{-3}\). This skill allows for cleaner math in fields ranging from engineering to finance.