Marta takes a loan of \( \$ 12,450 \) at an interest rate of \( 4.56 \% \), compounded quarterly. Assuming no payments are made, find the amount owed after 5 years. Round to the nearest dollar. Type your numerical answer below.

To find the amount owed after 5 years, we can use the formula for compound interest: \[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \] where: - \( A \) is the amount owed after \( t \) years, - \( P \) is the principal amount (initial loan amount), - \( r \) is the annual interest rate, - \( n \) is the number of times the interest is compounded per year, - \( t \) is the time in years. Given: - \( P = \$12,450 \), - \( r = 4.56\% = 0.0456 \), - \( n = 4 \) (compounded quarterly), - \( t = 5 \) years. Substitute these values into the formula to find the amount owed after 5 years. Calculate the value by following steps: - step0: Calculate: \(12450\left(1+\frac{0.0456}{4}\right)^{4}\times 5\) - step1: Divide the terms: \(12450\left(1+\frac{57}{5000}\right)^{4}\times 5\) - step2: Add the numbers: \(12450\left(\frac{5057}{5000}\right)^{4}\times 5\) - step3: Multiply the terms: \(62250\left(\frac{5057}{5000}\right)^{4}\) - step4: Simplify: \(62250\times \frac{5057^{4}}{5000^{4}}\) - step5: Rewrite the expression: \(250\times 249\times \frac{5057^{4}}{5000^{4}}\) - step6: Rewrite the expression: \(250\times 249\times \frac{5057^{4}}{250^{4}\times 20^{4}}\) - step7: Reduce the numbers: \(249\times \frac{5057^{4}}{250^{3}\times 20^{4}}\) - step8: Multiply the fractions: \(\frac{249\times 5057^{4}}{250^{3}\times 20^{4}}\) The amount owed after 5 years is approximately $65,137.51. Rounding to the nearest dollar, the amount owed is $65,138.