10 Amir invests \( \$ 1500 \) in an account. The account pays compound interest at a rate of \( r \% \) per year. At the end of 8 years the value of his investment is \( \$ 1656.73 \) Find the value of \( r \).

Given: - Initial investment: $1500 - Final value of the investment after 8 years: $1656.73 - Compound interest rate: \( r \% \) per year We can use the formula for compound interest to find the value of \( r \): \[ A = P \left(1 + \frac{r}{100}\right)^n \] where: - \( A \) is the final amount - \( P \) is the principal amount (initial investment) - \( r \) is the annual interest rate - \( n \) is the number of years Substitute the given values into the formula: \[ 1656.73 = 1500 \left(1 + \frac{r}{100}\right)^8 \] Now, we need to solve for \( r \). Let's use the function to solve the equation. Solve the equation by following steps: - step0: Solve for \(r\): \(1656.73=1500\left(1+\frac{r}{100}\right)^{8}\) - step1: Swap the sides: \(1500\left(1+\frac{r}{100}\right)^{8}=1656.73\) - step2: Convert the expressions: \(1500\left(1+\frac{r}{100}\right)^{8}=\frac{165673}{100}\) - step3: Multiply by the reciprocal: \(1500\left(1+\frac{r}{100}\right)^{8}\times \frac{1}{1500}=\frac{165673}{100}\times \frac{1}{1500}\) - step4: Multiply: \(\left(1+\frac{r}{100}\right)^{8}=\frac{165673}{150000}\) - step5: Simplify the expression: \(1+\frac{r}{100}=\pm \sqrt[8]{\frac{165673}{150000}}\) - step6: Simplify: \(1+\frac{r}{100}=\pm \frac{\sqrt[8]{165673\times 150000^{7}}}{150000}\) - step7: Separate into possible cases: \(\begin{align}&1+\frac{r}{100}=\frac{\sqrt[8]{165673\times 150000^{7}}}{150000}\\&1+\frac{r}{100}=-\frac{\sqrt[8]{165673\times 150000^{7}}}{150000}\end{align}\) - step8: Calculate: \(\begin{align}&r=\frac{\sqrt[8]{165673\times 150000^{7}}-150000}{1500}\\&r=-\frac{\sqrt[8]{165673\times 150000^{7}}+150000}{1500}\end{align}\) - step9: Rewrite: \(r_{1}=-\frac{\sqrt[8]{165673\times 150000^{7}}+150000}{1500},r_{2}=\frac{\sqrt[8]{165673\times 150000^{7}}-150000}{1500}\) The solutions for \( r \) are: \[ r_{1} = -\frac{\sqrt[8]{165673 \times 150000^{7}} + 150000}{1500} \] \[ r_{2} = \frac{\sqrt[8]{165673 \times 150000^{7}} - 150000}{1500} \] Since the interest rate cannot be negative, the value of \( r \) is: \[ r = \frac{\sqrt[8]{165673 \times 150000^{7}} - 150000}{1500} \] Let's calculate the value of \( r \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(\sqrt[8]{165673\times 150000^{7}}-150000\right)}{1500}\) - step1: Remove the parentheses: \(\frac{\sqrt[8]{165673\times 150000^{7}}-150000}{1500}\) The value of the interest rate \( r \) is approximately 1.250006%. Therefore, the value of \( r \) is 1.250006%.