A new car is purchased for 19900 dollars. The value of the car depreciates at \( 7.5 \% \) per year. What will the value of the car be, to the nearest cent, after 8 years?

To find the value of the car after 8 years, we can use the formula for depreciation: \[ \text{Value after depreciation} = \text{Initial value} \times (1 - \text{Depreciation rate})^{\text{Number of years}} \] Given: - Initial value of the car = $19900 - Depreciation rate = 7.5% per year - Number of years = 8 Substitute the values into the formula: \[ \text{Value after depreciation} = 19900 \times (1 - 0.075)^8 \] Now, we can calculate the value of the car after 8 years. Calculate the value by following steps: - step0: Calculate: \(19900\left(1-0.075\right)^{8}\) - step1: Subtract the numbers: \(19900\times 0.925^{8}\) - step2: Convert the expressions: \(19900\left(\frac{37}{40}\right)^{8}\) - step3: Simplify: \(19900\times \frac{37^{8}}{40^{8}}\) - step4: Rewrite the expression: \(20\times 995\times \frac{37^{8}}{40^{8}}\) - step5: Rewrite the expression: \(20\times 995\times \frac{37^{8}}{20^{8}\times 2^{8}}\) - step6: Reduce the numbers: \(995\times \frac{37^{8}}{20^{7}\times 2^{8}}\) - step7: Rewrite the expression: \(5\times 199\times \frac{37^{8}}{20^{7}\times 2^{8}}\) - step8: Rewrite the expression: \(5\times 199\times \frac{37^{8}}{5^{7}\times 4^{7}\times 2^{8}}\) - step9: Reduce the numbers: \(199\times \frac{37^{8}}{5^{6}\times 4^{7}\times 2^{8}}\) - step10: Multiply: \(\frac{199\times 37^{8}}{65536000000}\) The value of the car after 8 years will be approximately $10665.64 to the nearest cent.