Find the amount Erica owes at the end of 6 years if $\$ 5000$ is loaned to her at a rate of $8 \%$ compounded monthly. Use $A=P\left(1+\frac{r}{n}\right)^{n t}$
The amount owed is $\$ \square$
(Do not round until the final answer. Then round to the nearest cent as needed.)

Step 1 ：Given that Erica was loaned $5000 at an 8% interest rate compounded monthly for 6 years, we need to find the amount she owes at the end of this period.

Step 2 ：We use the formula for compound interest, which is given as \(A=P\left(1+\frac{r}{n}\right)^{n t}\), where:

Step 3 ：\(A\) is the amount of money accumulated after n years, including interest.

Step 4 ：\(P\) is the principal amount (the initial amount of money).

Step 5 ：\(r\) is the annual interest rate (in decimal).

Step 6 ：\(n\) is the number of times that interest is compounded per year.

Step 7 ：\(t\) is the time the money is invested for in years.

Step 8 ：In this case, \(P = 5000\), \(r = 8%\) or \(0.08\), \(n = 12\) (since it's compounded monthly), and \(t = 6\) years. We can substitute these values into the formula to find the final amount Erica owes.

Step 9 ：Substituting the given values into the formula, we get \(A = 5000\left(1+\frac{0.08}{12}\right)^{12 \times 6}\)

Step 10 ：Solving the above expression, we find that \(A = 8067.510836549617\)

Step 11 ：Rounding to the nearest cent, we get \(A = 8067.51\)

Step 12 ：Final Answer: The amount Erica owes at the end of 6 years is \(\boxed{\$8067.51}\)