Exponent Rules and Compound Interest Notes

Key idea from transcript: when dealing with powers with the same base, you combine exponents according to simple rules.
Exponent rules (with same base):
- Multiplication of like bases: if the bases are the same, you add the exponents.
- Formula: a^{m} \cdot a^{n} = a^{m+n}
- Intuition: you are multiplying the base by itself m+n times.
- Division of like bases: if you divide, you subtract the exponents.
- Formula: \frac{a^{m}}{a^{n}} = a^{m-n}
- If m < n, the result has a negative exponent (equivalently, it is the reciprocal of the positive-exponent form).
  - Example: \frac{a^{m}}{a^{n}} = a^{m-n} = \frac{1}{a^{n-m}}\quad (m<n)
- Intuition: removing factors of the base reduces the exponent.
The transcript emphasizes the relation between base and exponent and notes:
- When the bases are the same, exponents can be added (multiplication).
- When dividing, the exponent difference is used, and negative exponents arise if the exponent on the numerator is smaller than the one in the denominator.
- The exponent represents how many times the base is used as a factor; division moves some of those factors to the denominator (leading to negative exponents in the combined expression).
Example applications (from transcript statements):
- Example 1: 3^{2} \cdot 3^{5} = 3^{7}
- Example 2: 2^{8} / 2^{3} = 2^{5}
- Example 3 (negative exponent): 5^{-2} = \dfrac{1}{5^{2}} = 0.04
Transition to financial context: exponent rules are later applied to compound interest formulas, where the exponent represents the total number of compounding events.

Compound Interest: General Formula and Variations

Context from transcript: the exponent in the compound formula reflects the number of compounding terms per year, n. The rate is often expressed as APR, and the per-period rate is APR divided by the number of compounding periods per year.
Key terms:
- P = principal (initial amount)
- APR = annual percentage rate (as a decimal or percent, convert to decimal for calculations)
- n = number of compounding periods per year (e.g., 12 for monthly, 4 for quarterly, 365 for daily)
- t = number of years
- A = amount after t years
Per-period rate and total periods:
- Per-period rate = (\dfrac{r}{n}) where (r) is the annual rate in decimal form (or (\dfrac{APR}{n}) if using APR as a decimal).
- Total number of compounding periods = (nt).
General compound interest formula:
- A = P \left( 1 + \frac{r}{n} \right)^{nt}
- Here, (r) is the annual rate (as decimal). If you start from APR, ensure it is converted to a decimal form for (r).
Specific case (monthly compounding): replace (n) with 12:
- A = P \left( 1 + \frac{r}{12} \right)^{12t}
- If using APR directly, you can write with APR as decimal: A = P \left( 1 + \frac{APR}{12} \right)^{12t}
Generalization: the same structure holds for any compounding frequency by choosing different (n) (e.g., daily with (n=365), quarterly with (n=4), etc.).
Continuous compounding (limit case):
- Formula: A = P e^{rt}
- Concept: as the number of compounding periods per year (n) increases without bound, the discrete formula approaches the continuous form.
- Relationship to the discrete formula: \lim_{n\to\infty} P \left(1 + \frac{r}{n}\right)^{nt} = P e^{rt}
Practical interpretation and implications:
- Increasing the compounding frequency (larger (n)) generally increases the amount (A) for the same principal, rate, and time, but returns diminish as (n) grows large.
- Continuous compounding represents an idealized limit; real-world compounding uses finite frequencies (monthly, daily, etc.).
Worked example (illustrative):
- Suppose (P = 1000), annual rate (r = 0.06) (6%), time (t = 5) years, and monthly compounding ((n = 12)).
- Calculation:
- Per-period rate: (\dfrac{r}{n} = \dfrac{0.06}{12} = 0.005)
- Total periods: (nt = 12 \times 5 = 60)
- Amount: A = 1000 \left(1 + 0.005\right)^{60} \approx 1000 \times 1.349 = \$1349.86\n
- For comparison, continuous compounding with the same parameters:
- A = 1000 e^{(0.06)(5)} = 1000 e^{0.30} \approx 1000 \times 1.34986 = \$1349.86
- Note: the monthly compounding result is very close to the continuous result in this example.
Connections to broader concepts:
- Exponential growth: the factor (\left(1 + \frac{r}{n}\right)^{nt}) grows exponentially with (t).
- Time value of money: money available now is worth more than the same amount in the future due to earning potential via interest.
- The exponent (nt) encodes the total number of times interest is applied; more frequent application accelerates growth.

Summary of Key Takeaways

When multiplying powers with the same base, add exponents: a^{m} \cdot a^{n} = a^{m+n}.
When dividing powers with the same base, subtract exponents: \frac{a^{m}}{a^{n}} = a^{m-n}.
Negative exponents correspond to reciprocals: a^{-k} = \frac{1}{a^{k}}.
In finance, the exponent in the compound interest formula counts the total number of compounding events: A = P \left(1 + \frac{r}{n}\right)^{nt}.
For monthly compounding, set (n = 12): A = P \left(1 + \frac{r}{12}\right)^{12t}.$
Continuous compounding uses the limit of the discrete formula as (n \to \infty): A = P e^{rt}. The discrete and continuous formulas converge, illustrating the relationship between finite and continuous compounding.
Practical takeaway: more frequent compounding increases returns, but the difference between high frequencies and continuous compounding is bounded; practical choices depend on product terms and fees.