If \( \$ 550 \) are deposited into an account with a \( 7.5 \% \) interest rate, compounded monthly, what is the balance after 14 years? \( F=\$[?] \) \( F=P\left(1+\frac{r}{n}\right)^{n t} \) Round to the nearest cent.

To find the balance after 14 years, we can use the formula for compound interest: \[ F = P\left(1 + \frac{r}{n}\right)^{nt} \] Given: - \( P = \$550 \) (initial deposit) - \( r = 7.5\% \) (interest rate) - \( n = 12 \) (compounded monthly) - \( t = 14 \) years Substitute the given values into the formula: \[ F = 550\left(1 + \frac{0.075}{12}\right)^{12 \times 14} \] Now, we can calculate the balance after 14 years. Calculate the value by following steps: - step0: Calculate: \(550\left(1+\frac{0.075}{12}\right)^{12\times 14}\) - step1: Divide the terms: \(550\left(1+\frac{1}{160}\right)^{12\times 14}\) - step2: Add the numbers: \(550\left(\frac{161}{160}\right)^{12\times 14}\) - step3: Multiply the numbers: \(550\left(\frac{161}{160}\right)^{168}\) - step4: Simplify: \(550\times \frac{161^{168}}{160^{168}}\) - step5: Rewrite the expression: \(10\times 55\times \frac{161^{168}}{160^{168}}\) - step6: Rewrite the expression: \(10\times 55\times \frac{161^{168}}{10^{168}\times 16^{168}}\) - step7: Reduce the numbers: \(55\times \frac{161^{168}}{10^{167}\times 16^{168}}\) - step8: Rewrite the expression: \(5\times 11\times \frac{161^{168}}{10^{167}\times 16^{168}}\) - step9: Rewrite the expression: \(5\times 11\times \frac{161^{168}}{5^{167}\times 2^{167}\times 16^{168}}\) - step10: Reduce the numbers: \(11\times \frac{161^{168}}{5^{166}\times 2^{167}\times 16^{168}}\) - step11: Multiply the fractions: \(\frac{11\times 161^{168}}{5^{166}\times 2^{839}}\) The balance after 14 years is approximately $1566.58. Therefore, the balance after 14 years is $1566.58.