Question

lesson 6: exit ticket
half interested or more interesting
based on the work from this task, explain what each of the following expression would mean in terms of money:
a. ( f(x)=1,245(1.08)^{4} cdot(1.08)^{3} )
b. ( f(x)=1,245(1.08)^{-2} )
c. ( f(x)=1,245((1.08)^{\frac{1}{3}})^{2} )

Explanation:

• For \(a\). \(f(x)=1,245(1.08)^{4}\cdot(1.08)^{3}\):
• Using the rule of exponents \(a^{m}\cdot a^{n}=a^{m + n}\), we can rewrite \((1.08)^{4}\cdot(1.08)^{3}\) as \((1.08)^{4 + 3}=(1.08)^{7}\). In terms of money, if we assume the initial amount is \(P = 1245\) and the growth factor is \(1.08\) (representing an \(8\%\) increase), this formula represents the amount of money after \(7\) compounding periods (e.g., years if compounded annually).
• For \(b\). \(f(x)=1,245(1.08)^{- 2}\):
• Using the rule \(a^{-n}=\frac{1}{a^{n}}\), so \(f(x)=\frac{1245}{(1.08)^{2}}\). In terms of money, if we assume the initial amount is \(P = 1245\) and the growth factor is \(1.08\), this formula represents the amount of money \(2\) periods in the past. It could be used to find the present - value - like amount if we know the future amount and the growth rate and we want to go back in time.
• For \(c\). \(f(x)=1,245((1.08)^{\frac{1}{3}})^{2}\):
• Using the rule \((a^{m})^{n}=a^{mn}\), we can rewrite \(((1.08)^{\frac{1}{3}})^{2}=(1.08)^{\frac{2}{3}}\). If we assume the initial amount is \(P = 1245\) and the growth factor for a full - period is \(1.08\), this formula represents the amount of money after \(\frac{2}{3}\) of a compounding period. For example, if the compounding period is a year, it could represent the amount of money after \(8\) months (since \(\frac{2}{3}\) of a year is \(8\) months) if the interest is compounded annually and we use a simple - minded extension of the compound - interest formula for partial periods.

Answer:

• \(a\). Represents the amount of money after \(7\) compounding periods (with an \(8\%\) growth rate per period) starting from an initial amount of \(1245\).
• \(b\). Represents the amount of money \(2\) periods in the past (with an \(8\%\) growth rate per period) from a future amount of \(1245\).
• \(c\). Represents the amount of money after \(\frac{2}{3}\) of a compounding period (with an \(8\%\) growth rate per full - period) starting from an initial amount of \(1245\).