If \( \$ 500 \) are deposited into an account with a \( 4.5 \% \) interest rate, compounded annually, what is the balance after 6 years? \[ F=\$[\text { ? ] } \] \( F=P\left(1+\frac{r}{n}\right)^{\text {nt }} \)

To find the balance after 6 years, we can use the formula for compound interest: \[ F = P\left(1 + \frac{r}{n}\right)^{nt} \] Given: - Principal amount (\( P \)) = $500 - Interest rate (\( r \)) = 4.5% = 0.045 - Compounding frequency (\( n \)) = 1 (compounded annually) - Time (\( t \)) = 6 years Substitute the given values into the formula: \[ F = 500\left(1 + \frac{0.045}{1}\right)^{1 \times 6} \] Now, we can calculate the balance after 6 years. Calculate the value by following steps: - step0: Calculate: \(500\left(1+0.045\right)^{6}\) - step1: Add the numbers: \(500\times 1.045^{6}\) - step2: Convert the expressions: \(500\left(\frac{209}{200}\right)^{6}\) - step3: Simplify: \(500\times \frac{209^{6}}{200^{6}}\) - step4: Rewrite the expression: \(100\times 5\times \frac{209^{6}}{200^{6}}\) - step5: Rewrite the expression: \(100\times 5\times \frac{209^{6}}{100^{6}\times 2^{6}}\) - step6: Reduce the numbers: \(5\times \frac{209^{6}}{100^{5}\times 2^{6}}\) - step7: Rewrite the expression: \(5\times \frac{209^{6}}{25^{5}\times 4^{5}\times 2^{6}}\) - step8: Rewrite the expression: \(5\times \frac{209^{6}}{5^{10}\times 4^{5}\times 2^{6}}\) - step9: Reduce the numbers: \(1\times \frac{209^{6}}{5^{9}\times 4^{5}\times 2^{6}}\) - step10: Multiply the fractions: \(\frac{209^{6}}{2^{16}\times 5^{9}}\) - step11: Evaluate the power: \(\frac{209^{6}}{65536\times 5^{9}}\) The balance after 6 years is approximately $651.13.