Exponential Expressions and Growth/Decay Thinking

1) Exponential expressions: base, exponent, and evaluation

An exponential expression is a compact way to write repeated multiplication. It has the form a^n, read “a to the n.”

• Base (a): the factor being multiplied repeatedly.
• Exponent (n): how many times the base is used as a factor.

For whole-number exponents (n = 1, 2, 3, ...), the meaning is:

a^n means multiply a by itself n times.

Evaluate expressions with whole-number exponents

Step-by-step method:

• Write the repeated multiplication.
• Multiply in order (or group factors if it helps).

Examples:

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• 3^4 = 3·3·3·3 = 81
• 10^3 = 10·10·10 = 1000
• 2^6 = 2·2·2·2·2·2 = 64

Be careful: exponents apply to the base only.

• -(2^3) = -8 because 2^3 = 8 first, then the negative sign applies.
• (-2)^3 = -8 because the base is -2 multiplied three times: (-2)(-2)(-2).
• (-2)^4 = 16 because there are an even number of negative factors.

Translate between repeated multiplication and exponent form

To convert repeated multiplication to exponent form:

• Identify the repeated factor (that’s the base).
• Count how many times it appears (that’s the exponent).

Examples:

• 5·5·5·5·5 = 5^5
• 0.8·0.8·0.8 = 0.8^3
• (-3)(-3)(-3)(-3) = (-3)^4

To convert exponent form to repeated multiplication:

• Write the base as a factor repeated n times.

Examples:

• 7^2 = 7·7
• 1.2^4 = 1.2·1.2·1.2·1.2

2) Exponents as a way to describe multiplicative change

Exponential expressions naturally model situations where a quantity changes by the same multiplicative factor each period (day, month, year, step, etc.).

If a quantity starts at P and each period it is multiplied by a factor r, then after n periods:

amount after n periods = P · r^n

Interpretation of the factor r:

• Growth: r > 1 (multiplies by more than 1 each period)
• Decay: 0 < r < 1 (multiplies by a fraction each period)

Example A: “multiplies by 1.05 each period” (5% growth)

Suppose a population is P = 2000 and it multiplies by 1.05 each year. After n years:

P(n) = 2000 · 1.05^n

Compute a few values step-by-step:

• After 1 year: 2000·1.05 = 2100
• After 2 years: 2000·1.05^2 = 2000·(1.05·1.05) = 2000·1.1025 = 2205

Notice the key idea: the increase each year is not a fixed number; it depends on the current amount.

Example B: “keeps 0.92 each period” (8% decay)

A device retains 92% of its value each year. Factor r = 0.92. If the initial value is $500:

V(n) = 500 · 0.92^n

After 3 years:

V(3) = 500 · 0.92^3 = 500 · (0.92·0.92·0.92) ≈ 500 · 0.778688 ≈ 389.34

Turning percent change into a factor

• Increase by 5% → multiply by 1 + 0.05 = 1.05
• Decrease by 12% → multiply by 1 − 0.12 = 0.88
• Increase by 40% → multiply by 1.40

3) Additive vs. multiplicative patterns (comparison exercises)

Many mistakes come from confusing “add the same amount each step” with “multiply by the same factor each step.” Use this checklist:

• Additive pattern: constant difference (e.g., +3 each step). Values change linearly.
• Multiplicative pattern: constant ratio (e.g., ×1.2 each step). Values change exponentially.

Mini-exercise 1: Identify the pattern

For each sequence, decide whether it is additive or multiplicative, and state the rule.

• 4, 7, 10, 13, 16, ... (differences: +3) → additive rule: “add 3”
• 5, 10, 20, 40, 80, ... (ratios: ×2) → multiplicative rule: “multiply by 2”
• 100, 90, 81, 72.9, ... (ratios: ×0.9) → multiplicative rule: “multiply by 0.9”
• 2, 6, 10, 14, 18, ... (differences: +4) → additive rule: “add 4”

Mini-exercise 2: Same first step, different long-term behavior

Two savings plans both start at $100.

• Plan A: add $10 each month.
• Plan B: multiply by 1.10 each month.

Compute month 1 and month 2:

• Plan A: 100 → 110 → 120
• Plan B: 100 → 110 → 121

Even though both reach 110 after one month, the multiplicative plan starts pulling ahead because it grows from the updated amount each time.

Mini-exercise 3: Decide which model fits

Choose additive or multiplicative.

• A tank loses 3 liters every hour → additive (subtract 3 each hour)
• A bacteria culture doubles every hour → multiplicative (×2 each hour)
• A coupon reduces the price by 20% → multiplicative (×0.8)
• A phone plan charges a base fee plus $15 per month → additive (adds a constant each month)

4) Build tables from expressions like a^n and interpret growth/decay

Tables help you see how exponential expressions behave as the exponent increases. A simple approach:

• Pick a base a.
• Compute a^n for n = 0, 1, 2, 3, ... (if you include n=0, remember it represents the starting point in many contexts).
• Look for the multiplicative factor between consecutive rows.

Table A: Growth when a > 1

Let a = 2.

As n increases, 2^n grows quickly because each step multiplies by 2.

Table B: Decay when 0 < a < 1

Let a = 0.8.

As n increases, 0.8^n shrinks toward 0 because each step keeps only 80% of the previous amount.

Table C: Include an initial amount P

Suppose an account starts at P = 300 and grows by a factor of 1.05 each period:

A(n) = 300 · 1.05^n

Interpretation: the multiplier 1.05 is applied repeatedly, so the amount increases by a larger number each period.

5) Quick checks: reading and writing exponential expressions

Quick Check A: Identify base and exponent

• In 7^3, base = 7, exponent = 3
• In (-4)^2, base = -4, exponent = 2
• In 0.9^5, base = 0.9, exponent = 5

Quick Check B: Translate to repeated multiplication

• 6^4 = 6·6·6·6
• (-2)^3 = (-2)(-2)(-2)
• 1.1^2 = 1.1·1.1

Quick Check C: Translate to exponent form

• 9·9·9 = 9^3
• 0.5·0.5·0.5·0.5 = 0.5^4
• (-7)(-7) = (-7)^2

Quick Check D: Write an expression from a context

• “Starts at 80 and multiplies by 1.03 each week for n weeks” → 80·1.03^n
• “Starts at 1200 and keeps 95% each month for n months” → 1200·0.95^n
• “A quantity doubles each step for n steps, starting at P” → P·2^n

Quick Check E: Spot additive vs multiplicative wording

• “Increases by 7 each period” → additive (not exponential)
• “Increases by 7% each period” → multiplicative factor 1.07
• “Decreases by 15 each period” → additive
• “Decreases by 15% each period” → multiplicative factor 0.85