Brent Pickett borrowed \( \$ 8000 \) from his brother Dave. He agreed to repay the money at the end of 2 years, giving Dave the same amount of interest that he would have received if the money had been invested at \( 1.25 \% \) compounded quarterly. How much money did Brent repay his brother? Brent repaid his brother \( \$ \square \). (Round to the nearest cent as needed.)

To solve this problem, we need to calculate the amount of money Brent repaid his brother, which includes the principal amount borrowed and the interest earned. Given: - Principal amount borrowed = $8000 - Interest rate = 1.25% compounded quarterly - Time period = 2 years First, we need to calculate the interest earned if the money had been invested at 1.25% compounded quarterly for 2 years. The formula for compound interest is: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount. - \( r \) is the annual interest rate (in decimal form). - \( n \) is the number of times that interest is compounded per year. - \( t \) is the time the money is invested for in years. In this case, the interest rate is 1.25% or 0.0125 in decimal form, and the money is compounded quarterly, so \( n = 4 \) and \( t = 2 \) years. Substitute the values into the formula to calculate the interest earned: \[ A = 8000 \left(1 + \frac{0.0125}{4}\right)^{4 \times 2} \] Now, we can calculate the amount of money Brent repaid his brother by adding the principal amount and the interest earned. Calculate the value by following steps: - step0: Calculate: \(8000\left(1+\frac{0.0125}{4}\right)^{4\times 2}\) - step1: Divide the terms: \(8000\left(1+\frac{1}{320}\right)^{4\times 2}\) - step2: Add the numbers: \(8000\left(\frac{321}{320}\right)^{4\times 2}\) - step3: Multiply the numbers: \(8000\left(\frac{321}{320}\right)^{8}\) - step4: Simplify: \(8000\times \frac{321^{8}}{320^{8}}\) - step5: Rewrite the expression: \(320\times 25\times \frac{321^{8}}{320^{8}}\) - step6: Reduce the numbers: \(25\times \frac{321^{8}}{320^{7}}\) - step7: Rewrite the expression: \(25\times \frac{321^{8}}{5^{7}\times 64^{7}}\) - step8: Rewrite the expression: \(5^{2}\times \frac{321^{8}}{5^{7}\times 64^{7}}\) - step9: Reduce the numbers: \(1\times \frac{321^{8}}{5^{5}\times 64^{7}}\) - step10: Multiply the fractions: \(\frac{321^{8}}{3125\times 64^{7}}\) The interest earned if the money had been invested at 1.25% compounded quarterly for 2 years is approximately $8202.20. Therefore, Brent repaid his brother $8202.20.