Question

For the following exercises, use the compound interest formula, displaystyle{A}{left({t}right)}={P}{left({1}+frac{r}{{n}}right)}^{{{n}{t}}}. Use properties of rational exponents to solve the compound interest formula for the interest rate, r.

Rational exponents and radicals
For the following exercises, use the compound interest formula, $$\displaystyle{A}{\left({t}\right)}={P}{\left({1}+\frac{r}{{n}}\right)}^{{{n}{t}}}$$.
Use properties of rational exponents to solve the compound interest formula for the interest rate, r.

2021-01-29
Calculation:
We have the compound interest formula:
$$\displaystyle{A}{\left({t}\right)}={P}{\left({1}+\frac{r}{{n}}\right)}^{{{n}{t}}}$$
where A(t) is the account value, t is measured in years, P is the starting amount of the account, often called the principal, or more generally present value, r is the annual percentage rate (APR) expressed as a decimal, and n is the number of compounding periods in one year.
Multiply both sides by the power of $$\displaystyle\frac{1}{{{n}{t}}},$$ we get
$$\displaystyle\frac{{A}^{1}}{{{n}{t}}}={P}{\left({1}+\frac{r}{{n}}\right)}$$
Divide both sides by P
$$\displaystyle\frac{{{A}^{{\frac{1}{{{n}{t}}}}}}}{{P}}={\left({1}+\frac{r}{{n}}\right)}$$
Substract 1 from both sides
$$\displaystyle\frac{{{A}^{{\frac{1}{{{n}{t}}}}}}}{{P}}-{1}=\frac{r}{{n}}$$
Multiply both sides by n, we get
$$\displaystyle{n}{\left(\frac{{{A}^{{\frac{1}{{{n}{t}}}}}}}{{P}}-{1}\right)}={r}$$
Hence, $$\displaystyle{r}={n}{\left(\frac{{{A}^{{\frac{1}{{{n}{t}}}}}}}{{P}}-{1}\right)}$$