If (1.001)¹²⁵⁹ = 3.52 and (1.001)²⁰⁶² = 7.85, then (1.001)³³²¹ =

Exponents Problem: Calculating (1.001)³³²¹

This problem requires us to determine the value of a power, specifically $(1.001{)}^{3321}$, by leveraging two other given power values that share the same base. To approach this, we will use a fundamental property of exponents.

Understanding Exponent Properties

A crucial rule in algebra for exponents states that when you multiply two exponential terms that have the same base, you can add their exponents. This rule is often written as:

- $a^m\times a^n=a^{m+n}$

In our particular question, the base of the exponential terms is $1.001$.

Analyzing the Given Information

We are provided with the following values:

- The value of $(1.001{)}^{1259}$ is $3.52$.
- The value of $(1.001{)}^{2062}$ is $7.85$.

Our objective is to find the value of $(1.001{)}^{3321}$. Let's examine if there's a connection between the target exponent $3321$ and the exponents given in the problem, $1259$ and $2062$. If we add these two exponents:

- $1259+2062=3321$

This sum matches the exponent we need to calculate, which confirms we can use the multiplication rule for exponents.

Applying the Exponent Rule for Calculation

Since $3321$ is the sum of $1259$ and $2062$, we can express $(1.001{)}^{3321}$ as a product of two powers:

- First, rewrite the expression: $(1.001{)}^{3321}=(1.001{)}^{1259+2062}$
- Next, apply the exponent rule $a^{m+n}=a^m\times a^n$:
- $(1.001{)}^{1259+2062}=(1.001{)}^{1259}\times (1.001{)}^{2062}$

Now, substitute the numerical values given in the problem into this equation:

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

Performing the Numerical Multiplication

To get the final answer, we multiply the two decimal numbers:

- $3.52\times 7.85=27.632$

The calculated value is $27.632$.

Matching with Options

When comparing our result of $27.632$ with the provided options, we look for the closest value. It's common in these types of problems for intermediate values or final answers to be rounded slightly. The closest option to $27.632$ is $27.64$.

Therefore, $(1.001{)}^{3321}$ is approximately $27.64$.