To solve the equation \(\frac{1231}{390} = \left(1 + \frac{r}{100}\right)^{18}\), follow these steps:

### Step 1: Simplify the left-hand side
Calculate the division:

\[
\frac{1231}{390} \approx 3.1577
\]

So the equation becomes:

\[
3.1577 = \left(1 + \frac{r}{100}\right)^{18}
\]

### Step 2: Take the 18th root of both sides
To isolate \(\left(1 + \frac{r}{100}\right)\), take the 18th root of both sides:

\[
\left(3.1577\right)^{\frac{1}{18}} = 1 + \frac{r}{100}
\]

### Step 3: Compute the 18th root
Now, calculate the 18th root of 3.1577:

\[
3.1577^{\frac{1}{18}} \approx 1.0663
\]

So:

\[
1.0663 = 1 + \frac{r}{100}
\]

### Step 4: Solve for \(r\)
Subtract 1 from both sides to isolate \(\frac{r}{100}\):

\[
\frac{r}{100} = 0.0663
\]

Multiply both sides by 100 to solve for \(r\):

\[
r \approx 6.63\%
\]

### Final Answer:
The value of \(r\) is approximately **6.63%**.

Would you like more details or have any other questions?

Here are some related questions:
1. How can you solve exponential equations with different bases?
2. What is the significance of taking roots in exponential equations?
3. How do you simplify an equation involving a compound interest formula?
4. Can logarithms be used to solve this type of equation?
5. How does changing the value of \(r\) affect the growth rate in exponential equations?

**Tip:** When solving exponential equations, using logarithms can often simplify the process, especially when the exponent is not an integer.

Learn how to solve the exponential equation 1231 divided by 390 equals (1 plus r divided by 100) raised to the power of 18. This problem involves compound interest and exponential equation solving. Suitable for advanced high school and college-level students.

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