Solving the Exponential Equation $(1.001)^{3321}$

The problem asks us to find the value of $(1.001)^{3321}$ given two other values involving the same base raised to different powers.

Key Exponent Property

We can use the exponent rule: $a^{m+n} = a^m \times a^n$.

Applying the Property

1. Observe the relationship between the exponents: $1259 + 2062 = 3321$.

2. Therefore, we can rewrite $(1.001)^{3321}$ using the exponent rule:

   $ (1.001)^{3321} = (1.001)^{1259 + 2062} = (1.001)^{1259} \times (1.001)^{2062} $

3. Substitute the given values into the equation:

   $ (1.001)^{3321} = 3.52 \times 7.85 $

4. Perform the multiplication:

   $ 3.52 \times 7.85 = 27.632 $

5. The calculated value $27.632$ is approximately equal to the option $27.64$.

Final Answer Derivation

Based on the calculation using the exponent property $a^{m+n} = a^m \times a^n$, the value of $(1.001)^{3321}$ is found by multiplying the given values $(1.001)^{1259} = 3.52$ and $(1.001)^{2062} = 7.85$. The result is $3.52 \times 7.85 = 27.632$, which matches option 4.