Use the exponential rules to solve equations

• In any equation, if the unknown is inside a power, you can use roots (e.g. square roots) to remove the power.
• Remember to follow the rules and principles of algebra.

For example:

James invested \($100\) in an account. After \(20\) time periods, the investment returned \($180\). What was the interest rate per period for the investment.

We need to use the compound interest formula \[ A = P \left(1+\frac{r}{100}\right)^n\]  where \(A \) is the investment return, \(P\) is the amount invested, \(n\) is the number of time periods, and \(r\) is the interest rate. Substituting the values into the formula gives:

\(
\newcommand{\eqncomment}[2]{\small{\text{ #2}} } 
\newcommand{\ceqns}{\begin{array}{rcll}}
\newcommand{\ceqne}{\end{array}}
\)
\[ 
\ceqns 
180 &=& 100 \left(1+\frac{r}{100}\right)^{20} \\  
\frac{180}{100} &=& \left(1+\frac{r}{100}\right)^{20} & \eqncomment{0.3}{dividing by 100} \\  
1.8 &=& \left(1+\frac{r}{100}\right)^{20} \\  
(1.8)^{\frac{1}{20}}&=& \left[ \left(1+\frac{r}{100}\right)^{20} \right]^{\frac{1}{20}} & \eqncomment{0.3}{taking the 20th root} \\  
\sqrt[20]{1.8} &=& 1+\frac{r}{100} \\  
\sqrt[20]{1.8} -1 &=& \frac{r}{100} \\  
(\sqrt[20]{1.8}-1) \times 100&=& r \\  
r &\approx& 2.98   
\ceqne
\]

Therefore, the interest rate per period is approximately \(2.98\%\).