The formula for continuously compounded interest is given by $$ A=P e^{r t} $$, where $$ A $$ is the account balance, $$ P $$ is the principal, $$ r $$ is the annual interest rate (expressed as a decimal), and $$ t $$ is the time in years. If $$ \$ 5000 $$ is deposited into an account earning an annual interest rate of $$ 2.5 \% $$, compounded continuously, then what is the account balance after 8 years? Round your answer to the nearest dollar and do not include the $$ \$ $$ symbol in your answer.

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To find the account balance after 8 years with continuously compounded interest, we can use the formula for continuously compounded interest:

$$ A = P e^{r t} $$

Given:
- Principal ($$ P $$) = $5000
- Annual interest rate ($$ r $$) = 2.5% = 0.025
- Time ($$ t $$) = 8 years

Substitute the given values into the formula:

$$ A = 5000 e^{0.025 \times 8} $$

Now, we can calculate the account balance after 8 years using the formula.
Calculate the value by following steps:
- step0: Calculate:
  $$5000e^{0.025\times 8}$$
- step1: Multiply the numbers:
  $$5000e^{0.2}$$
- step2: Convert the expressions:
  $$5000e^{\frac{1}{5}}$$
- step3: Rewrite the expression:
  $$5000\sqrt[5]{e}$$
The account balance after 8 years with continuously compounded interest is approximately $6107.01.

Therefore, the account balance after 8 years is $6107.
The account balance after 8 years is $6107.

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