If $(1.001{)}^{1259}=3.52$ and $(1.001{)}^{2062}=7.85$, then $(1.001{)}^{3321}=$

**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.