Represent 6³ as an extended product. Do not evaluate.
Exponents represent repeated multiplication. Write each factor out fully, group like bases, then recount how many times each base is multiplied.
| (3x²)³(2x⁵)³ | Write each factor as an extended product |
| = 3·3·3·2·2·2 · x²·x²·x²·x⁵·x⁵·x⁵ | Group the numbers together and the x-factors together |
| = 216x²¹ | Multiply coefficients; add exponents on the like base |
| x⁵ / x⁹ | Split into (x⁵/x⁵) · (1/x⁴) |
| = 1 · 1/x⁴ = 1/x⁴ | The matching factors cancel to 1, leaving the extra factors in the denominator |
| (3x²)³ / 9x⁴ | Expand the numerator: 3x²·3x²·3x² = 27x⁶ |
| = 27x⁶ / 9x⁴ | Split into (9x⁴/9x⁴) · (3x²/1) |
| = 3x² | The matching piece cancels to 1 |
Exponent rules are shortcuts for counting factors. x³·x² = five x's multiplied → add exponents. (x³)² = three x's, twice → multiply exponents. Division cancels shared factors → subtract exponents. If you forget a rule, expand a tiny example and count.
| Simplify: (5x³)² · x⁴ | 25x⁶ · x⁴ = 25x¹⁰. Power rule first, then product rule. |
| Simplify: 18x⁷y² / (6x³y) | 3x⁴y. Divide coefficients, subtract exponents per base. |
Simplify: x⁵/x²
If positive exponents mean multiplying by the base repeatedly, negative exponents mean dividing by the base repeatedly. The pattern 2⁴=16, 2³=8, 2²=4, 2¹=2 keeps dividing by 2 each step down — so 2⁰=1, 2⁻¹=1/2, 2⁻²=1/4, and so on.
| b⁰ = 1 (b ≠ 0) | Zero Exponent Rule |
| b⁻ⁿ = 1/bⁿ | Negative Exponent Rule — flips the base to the denominator |
| f(x) = 3x⁻² + 2x⁰, find f(2) | Substitute x = 2 |
| = 3(2)⁻² + 2(2)⁰ = 3·(1/4) + 2(1) | Evaluate each exponent first |
| = 3/4 + 2 = 2¾ | Then multiply and add |
| 2⁴/2⁴ | Exponent rule: 2⁴⁻⁴ = 2⁰ = 1 |
| x²/x⁷ | Exponent rule: x²⁻⁷ = x⁻⁵ = 1/x⁵ — negative and zero exponents keep the subtraction rule consistent |
A zero exponent gives 1 because dividing x³ by x³ must equal 1 — and the subtraction rule says it's x⁰. A negative exponent means 'wrong side of the fraction bar': move the factor across the bar and the exponent turns positive. Negative exponents never make a result negative — they make it a reciprocal.
| Simplify: 4⁻² | = 1/4² = 1/16. Reciprocal, not negative. |
| Simplify: x⁻³y² / x² | = y²/x⁵. The x⁻³ drops below, joining x²: x³·x² = x⁵. |
Evaluate 2⁻³ without a calculator.
Many real quantities grow by being multiplied by the same factor over and over — a rumor doubling each day, a population increasing each hour. That's exponential growth.
| Rumor starts with 3 people, doubles each day | N = 3 · 2ᵈ, where d = number of days |
| N(20) = 3(2)²⁰ = 3,145,728 | Plug in d = 20 to predict people who know it after 20 days |
| Helmut starts 160 ft from a windmill, walks half the remaining distance each trip | D = 160(1/2)ⁿ, where n = number of trips |
| D(6) = 160(1/2)⁶ = 2.5 ft | He gets closer and closer but the base being less than 1 means he never actually reaches 0 |
In both examples, the input (days, trips) counts discrete events — you can't have "2.5 trips." So the domain of these models is {0, 1, 2, 3, ...}, not all real numbers.
Exponential change multiplies by the same factor each period instead of adding the same amount. The model y = a(b)t: a is the start, b the repeated multiplier. Growth has b > 1; decay has 0 < b < 1. From a percent: growth rate r gives b = 1 + r; decay gives b = 1 − r.
| $500 grows 6% per year. Value after 3 years? | 500(1.06)³ ≈ $595.51. Multiply by 1.06 three times — not 500 + 18%. |
| A 100 mg dose loses 20% per hour. Amount after 4 hours? | 100(0.8)⁴ = 40.96 mg. Keeping 80% each hour, four times. |
If N(d) = 3(2)ᵈ, find N(2).
y = a(b)ˣ, where a is the y-intercept (starting value, at x = 0) and b is the base or "growth factor" (what you multiply by for each step of x).
| f(x) = 8(2)ˣ | a = 8 (y-intercept), b = 2 (growth factor) |
| f(0) = 8, f(2) = 32, f(−1) = 4 | Plugging in x always multiplies the starting value by b that many times |
| y = a(b)ˣ increases if b > 1 | Each step multiplies by more than 1, so values grow |
| y = a(b)ˣ decreases if 0 < b < 1 | Each step multiplies by a fraction, so values shrink toward 0 |
| x: 0,1,2,3,4 y: 10,30,90,270,810 | y-intercept (a-value) is the output at x = 0, so a = 10 |
| y = 10(3)ˣ | Each y-value is 3× the previous one, so b = 3 |
Reading y = a(b)x: plug x = 0 and everything but a vanishes — so a is always the y-intercept/starting value. b answers 'what do I multiply by each step?' To find b from a table with consecutive inputs, divide any output by the previous one.
| Table x: 0,1,2 → y: 6, 18, 54. Write the function. | a = 6 (value at x = 0), b = 18/6 = 3 → y = 6(3)x. |
| For y = 40(0.5)x, what does the 0.5 mean? | The quantity halves each step — starting at 40: 20, 10, 5, … |
A shirt costs $30. Set up a ratio to find 15% of $30.
A percent always compares two quantities as a proportional relationship out of 100. You can always solve a percent problem by setting up p/100 = x/total.
| Jonathan's pay: $12.50 → $14.75. What percent increase? | 2.25/12.50 = x/100 → 12.50x = 225 |
| x = 18% | Divide both sides by 12.50 |
| 15% tip on a $35 meal | Tip = 35 × 0.15 = $5.25 |
| 8.5% of 250 | = 0.085 × 250 (NOT 0.85 or 8.5 — line up the decimal with the percent) |
| $45 jeans, 8% sales tax — total cost? | Tax = 0.08 × 45 = $3.60; Total = 45 + 3.60 = $48.60 |
Percent means 'per hundred' — every percent problem is the same triangle: part = percent × whole. Convert the percent to a decimal, identify which of the three pieces is missing, and solve. 'Of' means multiply; 'is' means equals.
| What is 35% of 80? | 0.35 × 80 = 28. |
| 18 is what percent of 45? | 18 = p × 45 → p = 0.40 = 40%. |
Increase 350 by 6% using a single multiplication (from Lesson 6).
If a quantity grows by a constant percent rate r each period, its model is P(t) = a(1 + r)ᵗ, where a is the starting amount and t is the number of periods.
| 28 fruit flies, growing 6% per hour | P(t) = 28(1.06)ᵗ |
| P(24) = 28(1.06)²⁴ ≈ 113 flies | Plug in the number of hours to predict the population |
| Pool starts at 12 ft, drains 20% per hour | D(t) = 12(0.80)ᵗ — always model with the percent that REMAINS |
| A(t) = 250(1.15)ᵗ, an oil spill's area in ft² after t days | A(0) = 250 ft² initial size; growing 15% per day (since b = 1.15) |
To build an exponential model from a percent: the multiplier is b = 1 ± r (plus for growth, minus for decay), so 'grows 3% yearly from 2,000' becomes y = 2000(1.03)t. Read models in reverse the same way: b = 1.07 means 7% growth; b = 0.94 means 6% decay.
| A town of 15,000 shrinks 2% per year. Model and 10-year prediction? | P = 15000(0.98)t → P(10) ≈ 12,268. |
| What percent change does y = 320(1.045)t describe? | 4.5% growth per time period — read the 0.045 above the 1. |
If P(t) = 282.2(1.009)ᵗ, evaluate P(1).
Linear functions are built on repeatedly ADDING the same amount (the slope). Exponential functions are built on repeatedly MULTIPLYING by the same amount (the base). Look at a table: does the y-value change by a constant difference (linear) or a constant factor (exponential)?
| Table 1: 5, 10, 20, 40, 80 (×2 each step) | Exponential — constant ratio → y = 5(2)ˣ |
| Table 2: 8, 11, 14, 17, 20 (+3 each step) | Linear — constant difference → y = 3x + 8 |
| Points (0, 12) and (1, 3) | Linear: b = 12 (y-int), m = (3−12)/1 = −9 → y = −9x + 12 |
| Same two points, exponential form | a = 12 (y-int), b = 3/12 = 1/4 → y = 12(1/4)ˣ |
A linear function's average rate of change is constant. An increasing exponential's average rate of change keeps growing. Even if the exponential starts out smaller and looks slower, it will eventually overtake any increasing linear function — it just takes more x-values to see it happen.
Linear vs exponential is 'add the same' vs 'multiply by the same.' In a table with evenly spaced inputs: constant differences → linear; constant ratios → exponential. Long-run: exponential growth eventually overtakes any linear function, no matter how steep the line starts.
| x: 1,2,3,4 → y: 3, 6, 12, 24 — which type? | Ratios are all 2 (differences are 3, 6, 12 — not constant): exponential, y = 1.5(2)x. |
| x: 1,2,3,4 → y: 10, 17, 24, 31 — which type? | Differences all +7: linear, y = 7x + 3. |
Is the table x: 0,1,2,3 → y: 2,10,50,250 linear or exponential?
Just as arithmetic sequences are discrete linear functions, geometric sequences are discrete exponential functions. Given a₁, each next term is aᵢ = aᵢ₋₁ · r, where r is the common ratio.
| 2, 6, 18, ... | r = 6/2 = 3; next terms: 54, 162 |
| 16, 8, 4, ... | r = 8/16 = 1/2; next terms: 2, 1 |
| b(1) = 3, b(n) = b(n−1)·2 | b(2)=3·2=6, b(3)=6·2=12, b(4)=12·2=24 |
| b(10) = 3·2⁹ = 1,536; b(20) = 3·2¹⁹ = 1,572,864 | General pattern: bₙ = a₁ · r ⁿ⁻¹ — count how many times you multiplied by r |
A geometric sequence is exponential growth in list form: each term is the previous times a common ratio r. Find r by dividing neighbors. Explicit formula an = a₁·rn−1: start at the first term, multiply by r once per step after it.
| Sequence 2, 6, 18, 54, … Find a⁶. | r = 3 → a⁶ = 2·3⁵ = 486. Five multiplications after the first term. |
| Is 40, 20, 10, 5 geometric? | Yes — r = ½. Geometric sequences can shrink; they just multiply by a fraction. |
Find 8% of 250 using a single multiplication.
To increase a number by p%, multiply it by (1 + p/100). The result already includes the original 100% plus the extra percent — no separate addition step needed.
| Increase 350 by 6% | New population = 350 + 350(0.06) = 350(1 + 0.06) |
| = 350(1.06) = 371 | Multiplying by 1.06 gives 106% of the original in one step |
| Decrease 200 by 8% | Percent remaining = 100% − 8% = 92% |
| = 200(0.92) = 184 | Multiply the original by the decimal form of what remains |
| $135 jacket, 35% off, then 7% sales tax | Apply the discount first as its own multiplication |
| After discount: 135(0.65) = $87.75 | 65% remains after a 35% discount |
| After tax: 87.75(1.07) = $93.89 | Tax is added onto the already-discounted price — a second, separate multiplication |
Increasing by 10% and then decreasing by 10% does NOT return you to the original number. Test it: 200 → 200(1.10) = 220 → 220(0.90) = 198. You land below 200, because the second change (10% of 220) is a bigger amount than the first (10% of 200).
Percent change always compares to the original: (new − original)/original. The one-step shortcut: increasing by 15% is multiplying by 1.15; decreasing by 15% is multiplying by 0.85. That multiplier idea is exactly what powers exponential models later.
| A $70 jacket is marked up 20%. New price? | 70 × 1.20 = $84 — one multiplication instead of computing 20% and adding. |
| A price fell from $80 to $68. Percent decrease? | (68 − 80)/80 = −0.15 → 15% decrease. |
These problems pull from earlier units on purpose. Switching between skills feels harder in the moment, but it helps you remember longer and matches how a real exam mixes topics. Try each one on paper first, then reveal.Estos problemas provienen de unidades anteriores a propósito. Cambiar de una destreza a otra se siente más difícil en el momento, pero te ayuda a recordar por más tiempo y se parece a cómo un examen real mezcla los temas. Intenta cada uno en papel primero, luego revela la respuesta.