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# Exponent Laws & Rules

### 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:

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

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