https://file.aiquickdraw.com/mathbot/file/1728566005771ad0j3hxw.jpg

To solve this compound interest problem, we will use the formula given in the image:

\[
F = P \left( 1 + \frac{r}{n} \right)^{nt}
\]

Where:
- $ F $ is the future balance (what we're solving for),
- $ P $ is the initial principal, which is $500,
- $ r $ is the annual interest rate, which is 4.5% (or 0.045 as a decimal),
- $ n $ is the number of times the interest is compounded per year (since it's compounded annually, $ n = 1 $),
- $ t $ is the time in years, which is 6.

### Step-by-step calculation:

1. The formula simplifies as the interest is compounded annually ($ n = 1 $):
\[
F = 500 \times \left( 1 + \frac{0.045}{1} \right)^{1 \times 6}
\]

2. This becomes:
\[
F = 500 \times \left( 1 + 0.045 \right)^{6}
\]
\[
F = 500 \times (1.045)^{6}
\]

3. Now, calculate $ (1.045)^6 $:
\[
1.045^6 \approx 1.296
\]

4. Multiply by the principal:
\[
F = 500 \times 1.296 = 648
\]

So, the balance after 6 years will be approximately **$648.00**.

Would you like a further breakdown or clarification on any step?

### Follow-up questions:
1. How would the balance change if the interest was compounded quarterly?
2. What would happen to the balance if the time period was extended to 10 years?
3. How does the future balance change if the interest rate is 5% instead of 4.5%?
4. What would be the balance after 6 years if $1000 was initially deposited?
5. How is compound interest different from simple interest in long-term savings?

**Tip:** Remember, compound interest grows exponentially over time, making a big difference in long-term investments compared to simple interest.

If $500 are deposited into an account with a 4.5% interest rate, compounded annually, what is the balance after 6 years?

Learn how to calculate the future balance of a $500 deposit with 4.5% annual interest compounded annually for 6 years. This problem covers key concepts in compound interest and exponential growth, using the formula F = P(1 + r/n)^(nt).

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