Exponentiation and Applications
Exponentiation is a mathematical operation, written as \(b^n\), involving two numbers, the base \(b\) and the exponent or power \(n\). When \(n\) is a positive integer, \(b^n\) represents the \(n\) times repeated multiplication of the base \(b\), that is
\[b^{n}=\underbrace{b\times \cdots \times b}_{n},\]
and is pronounced as “\(b\) raised to the power of \(n\)”.
In Excel, \(b^n\) is calculated by
the formula =b^n.
Make a table (see page 597 in the textbook) of the frequency of 2
octaves starting from middle C with a frequency of 260 cps. Use
=260*1.05946^N for \(N\)
half-steps over middle C. Your table may look like this:
| half steps above middle C | Note | Frequency |
|---|---|---|
| 0 | C | 260 |
| 1 | C# | 275.5 |
| 2 | D | 291.8 |
Solution: In a new worksheet, enter “half steps
above middle C” in the cell A1, “Note” in the cell
B1, and “Frequency” in the cell C1.
Enter 0, 1, 2 in the cells A2, A3 and
A4. Then select A2 to A4 and use
autofill to generate numbers up to 24.
Enter 260 in the cell C2 and the formula
=260*1.05946^(N) in C3. Then use autofill to
find frequencies for all notes within 2 octave above the middle C.
Enter the name of the notes in colum B and you will get
the table.
Make a table to represent the growth of your initial investment of
$500 at 4% interest compounded annually for 10 years. Use
=500*(1+0.04)^N for a formula. Your table may look like
this:
| Years | Balance in $ |
|---|---|
| 0 | 500.00 |
| 1 | 520.00 |
| 2 | 540.80 |
How long would it take for your investment to grow to $800 or more?
Solution: In a new worksheet, enter “Years” in the
cell A1 and “Balance(in $)”, in the cell
B1.
Enter 0, 1, 2 in the cells A2, A3 and
A4. Then select A2 to A4 and use
autofill to generate numbers up to 10.
Enter 500 in the cell B2 and the formula
=500*(1+0.04)^N in B3. Then use autofill to
create formulas for each \(n\)-th
year.
For the second part of the question, generate more rows until you see the balance near $800. You will see after 12 years, the balance will be over $800.
Remark: If one would like to apply for a loan $L,
with the annual interests \(r\)
compounded \(n\) times for t years,
then the period payment is given by the formula
=L(1+r/n)^(-n*t).
Make a table of the frequency of 2 octaves starting from one
octave below middle C with a frequency of 260 cps to one octaves above
middle C. Use =260*1.05946^N for \(N\) half-steps over middle C.
Hint: Put 0 for middle C in A13. Then -1 in A12, -2
in A11 and autofill (upward to A1). Put 1 in A14, 2 in A15 and then
autofill (downward) to A25. Then generate the frequency using the
formula =260*1.05946^N.
(Optional) Make a table to represent the growth
of your initial investment of $1000 at 4% interest compounded monthly
for 20 years. Use =500*(1+0.04/12)^(12*N) for a formula,
where \(N\) is the number of years.
How long would it take for your investment to grow to $10,000?