# 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.

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.

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, $$\displaystyle{r}={n}{\left(\frac{{{A}^{{\frac{1}{{{n}{t}}}}}}}{{P}}-{1}\right)}$$

### Relevant Questions

Solve the compound interest formula for the interest rate r using the properties of rational exponents. then use the obtained formula to calculate the interest rate for an account that was compounded semi-annually, had an initial deposit of $10,000 and was worth$14,373.53 after 11 years.
Rational exponents evaluate each expression.
(a) $$\displaystyle{27}^{{{1}\text{/}{3}}}$$
(b) $$\displaystyle{\left(-{8}\right)}^{{{1}\text{/}{3}}}$$
(c) $$\displaystyle-{\left(\frac{1}{{8}}\right)}^{{{1}\text{/}{3}}}$$
The values of x that satisfy the equation with rational exponents $$\displaystyle{\left({x}+{5}\right)}^{{\frac{3}{{2}}}}={8}$$ and check all the proposed solutions.
Express the radical as power.
$$\displaystyle{\left({a}\right)}{\sqrt[{{6}}]{{{x}^{5}}}}$$
To simplify:
The expression $$\displaystyle{\sqrt[{{6}}]{{{x}^{5}}}}$$ and express the answer using rational exponents.
(b) $$\displaystyle{\left(\sqrt{{x}}\right)}^{9}$$
To simplify:
The expression $$\displaystyle{\left(\sqrt{{x}}\right)}^{9}$$ and express the answer using rational exponents.
Use a right triangle to write the following expression as an algebraic expression. Assume that x is positive and in the domain of the given inverse trigonometric function.
Given:
$$\displaystyle \tan{{\left({{\cos}^{ -{{1}}}{5}}{x}\right)}}=?$$
a) Find the rational zeros and then the other zeros of the polynomial function $$\displaystyle{\left({x}\right)}={x}^{3}-{4}{x}^{2}+{2}{x}+{4},\ \tet{that is, solve}\ \displaystyle f{{\left({x}\right)}}={0}.$$
b) Factor $$f(x)$$ into linear factors.

Compute the following binomial probabilities directly from the formula for $$b(x, n, p)$$:

a) $$b(3,\ 8,\ 0.6)$$

b) $$b(5,\ 8,\ 0.6)$$

c) $$\displaystyle{P}{\left({3}≤{X}≤{5}\right)}$$

when $$n = 8$$ and $$p = 0.6$$

d)$$\displaystyle{P}{\left({1}≤{X}\right)}$$ when $$n = 12$$ and $$p = 0.1$$

The following quadratic function in general form, $$\displaystyle{S}{\left({t}\right)}={5.8}{t}^{2}—{81.2}{t}+{1200}$$ models the number of luxury home sales, S(t), in a major Canadian urban area, according to statistical data gathered over a 12 year period. Luxury home sales are defined in this market as sales of properties worth over \$3 Million (inflation adjusted). In this case, $$\displaystyle{\left\lbrace{t}\right\rbrace}={\left\lbrace{0}\right\rbrace}\ \text{represents}\ {2000}{\quad\text{and}\quad}{\left\lbrace{t}\right\rbrace}={\left\lbrace{11}\right\rbrace}$$represents 2011. Use a calculator to find the year when the smallest number of luxury home sales occurred. Without sketching the function, interpret the meaning of this function, on the given practical domain, in one well-expressed sentence.
$$\displaystyle{\sqrt[{{7}}]{{11}}}\times{\sqrt[{{6}}]{{13}}}$$
(a) As soon as the infusion of Taxol is completed, the drug concentration in a patient’s blood is 1000 nanograms per milliliter $$\displaystyle{\left(\frac{{{n}{g}}}{{{m}{l}}}\right)}.$$ 12 hours later there is $$\displaystyle{50}\frac{{{n}{g}}}{{{m}{l}}}$$ left in the patient’s system. Use the data to construct an appropriate formula modeling the blood concentration of Taxol as a function of time after the infusion is completed.
(b) As soon as the infusion of Abraxane is completed, the drug concentration in a patient’s blood is 1000 nanograms per milliliter $$\displaystyle{\left(\frac{{{n}{g}}}{{{m}{l}}}\right)}$$. 24 hours later there is $$\displaystyle{50}\frac{{{n}{g}}}{{{m}{l}}}$$ left in the patient’s system. Use the data to construct an appropriate formula modeling the blood concentration of Abraxane as a function of time after the infusion is completed.