Certainly! Let's break down the expression $ A = 5000(1 + 0.05)^3 $ step by step.

### Step 1: Understand the Expression
The expression represents the future value $A$ of an investment of $5,000 after 3 years, with a 5% annual interest rate compounded once per year.

- **5000**: This is the principal amount (initial investment).
- **$1 + 0.05$**: This represents the growth factor for one year. The $1$ represents the original principal, and $0.05$ is the 5% interest rate in decimal form.
- **3**: This is the number of years the money is invested.

### Step 2: Simplify the Growth Factor
Calculate the growth factor:

\[
1 + 0.05 = 1.05
\]

So the expression now becomes:

\[
A = 5000 \times 1.05^3
\]

### Step 3: Calculate the Exponent
Next, raise 1.05 to the power of 3 (since the investment is for 3 years):

\[
1.05^3 = 1.05 \times 1.05 \times 1.05
\]

Let's break it down:

\[
1.05 \times 1.05 = 1.1025
\]

Now multiply by 1.05 again:

\[
1.1025 \times 1.05 = 1.157625
\]

So, $1.05^3 = 1.157625$.

### Step 4: Multiply by the Principal
Finally, multiply the result by the principal amount:

\[
A = 5000 \times 1.157625 = 5788.125
\]

### Final Result
The value of the investment after 3 years is **$5,788.13**.

Would you like further details or have any questions?

Here are some related questions:
1. Can you explain how to calculate $1.05^3$ using logarithms?
2. How does the value change if the interest rate is 10%?
3. What happens if you invest $10,000 instead of $5,000?
4. How would the calculation change if the interest were compounded quarterly?
5. Can you show how to derive the compound interest formula from basic principles?
6. What is the impact of extending the investment to 4 years?
7. How does the formula apply if interest is compounded continuously?
8. What if the interest rate were variable? How would you calculate the future value?

**Tip:** Always ensure you convert the percentage rate to a decimal before using it in calculations.

Learn how to calculate the future value of an investment using compound interest with a step-by-step solution. This problem involves calculating the future value of $5,000 after 3 years with a 5% interest rate compounded annually. Suitable for high school students.

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