6.7Unit 6 Review
Comprehensive review of all Unit 6 topics: exponent rules, scientific notation, rational exponents and radicals, exponential functions, exponential growth and decay, and comparing linear and exponential functions.
Why This Matters
Exponents and exponential functions are foundational for Algebra 2, Precalculus, and AP Calculus. A strong review here also prepares you for logarithms — the inverse of exponential functions — which you'll encounter in every advanced math course.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How do the rules of exponents connect to exponential functions, and how can exponential models describe real-world change?
Unit 6 Overview
Unit 6 builds a complete toolkit for working with exponents and exponential functions. You started with the eight fundamental exponent rules, then extended them to very large and very small numbers using scientific notation. Rational exponents connected exponent rules to radical expressions. You studied exponential functions — first their general form f(x) = ab^x, then their real-world applications in growth and decay models. Finally, you compared linear and exponential functions to develop model-selection skills.
Exponent Rules
Scientific Notation
Rational Exponents & Radicals
Intro to Exponential Functions
Exponential Growth & Decay
Comparing Linear & Exponential
Chapter 1 — Exponent Rules
Exponent Rules — Quick Reference
| Rule | Formula | Example |
|---|---|---|
| Product Rule | aⁿ · aᵐ = aⁿ⁺ᵐ | 3² · 3⁴ = 3⁶ = 729 |
| Quotient Rule | aⁿ ÷ aᵐ = aⁿ⁻ᵐ | 5⁷ ÷ 5³ = 5⁴ = 625 |
| Power Rule | (aⁿ)ᵐ = aⁿᵐ | (2³)⁴ = 2¹² = 4096 |
| Zero Exponent | a⁰ = 1 (a ≠ 0) | 7⁰ = 1 |
| Negative Exponent | a⁻ⁿ = 1/aⁿ | 4⁻² = 1/16 |
| Power of Product | (ab)ⁿ = aⁿbⁿ | (2x)³ = 8x³ |
| Power of Quotient | (a/b)ⁿ = aⁿ/bⁿ | (3/5)² = 9/25 |
| Fractional Exponent | aᵐ/ⁿ = ⁿ√aᵐ | 8²/³ = ³√64 = 4 |
Chapter 2 — Scientific Notation
Scientific Notation — Conversion Guide
Standard Form → Scientific Notation
1. Move decimal so coefficient is between 1 and 10.
2. Count moves → that's the exponent of 10.
3. Left moves = positive exponent. Right moves = negative exponent.
| Standard Form | Scientific Notation | Note |
|---|---|---|
| 4,500,000 | 4.5 × 10⁶ | Move decimal 6 left → positive exponent |
| 0.000032 | 3.2 × 10⁻⁵ | Move decimal 5 right → negative exponent |
| 602,000,000,000,000,000,000,000 | 6.02 × 10²³ | Avogadro's number |
| 0.000000001 | 1 × 10⁻⁹ | 1 nanometer |
Chapter 3 — Rational Exponents & Radicals
Radical ↔ Rational Exponent Conversion
ⁿ√aᵐ = a^(m/n) ←→ a^(m/n) = ⁿ√aᵐ
Index of radical = denominator of exponent
| Radical Form | Rational Exponent | Example |
|---|---|---|
| √a | a^(1/2) | √25 = 25^(1/2) = 5 |
| ³√a | a^(1/3) | ³√27 = 27^(1/3) = 3 |
| ⁴√a | a^(1/4) | ⁴√16 = 16^(1/4) = 2 |
| ³√a² | a^(2/3) | ³√8² = 8^(2/3) = 4 |
| ⁵√a³ | a^(3/5) | ⁵√32³ = 32^(3/5) = 8 |
| ⁿ√aᵐ | a^(m/n) | General form: index n, power m |
Chapter 4 — Exponential Functions
An exponential function has the form f(x) = ab^x where a is the initial value and b is the base. The y-intercept is always (0, a). The horizontal asymptote is y = 0. When b > 1 the function grows; when 0 < b < 1 it decays.
f(x) = 2ˣ — Exponential Growth
f(x) = 32·(0.5)ˣ — Exponential Decay
Chapter 5 — Exponential Growth & Decay
Exponential Growth vs Decay — Side-by-Side
| Feature | Growth | Decay |
|---|---|---|
| General Form | f(x) = a(1 + r)ˣ | f(x) = a(1 − r)ˣ |
| Base (b) | b > 1 | 0 < b < 1 |
| Rate (r) | r > 0 (positive) | r > 0 (positive) |
| Output trend | Increases | Decreases |
| Graph shape | Rises steeply right | Falls toward y = 0 |
| Asymptote | y = 0 (below) | y = 0 (above) |
| Real-world ex. | Population, interest | Half-life, depreciation |
Compound Interest: A = P(1 + r/n)^(nt)
| Principal | Rate | Compounding | Years | Balance |
|---|---|---|---|---|
| $1,000 | 5% | Annually (1) | 10 | $1,628.89 |
| $1,000 | 5% | Quarterly (4) | 10 | $1,643.62 |
| $1,000 | 5% | Monthly (12) | 10 | $1,647.01 |
| $1,000 | 5% | Daily (365) | 10 | $1,648.66 |
| $5,000 | 3% | Monthly (12) | 20 | $9,070.09 |
| $2,500 | 7% | Quarterly (4) | 15 | $7,038.93 |
Half-Life Reference Table
Model: f(t) = a · (0.5)^(t / half-life)
| Substance | Half-Life | Application |
|---|---|---|
| Carbon-14 | 5,730 years | Archaeological dating |
| Iodine-131 | 8 days | Medical thyroid treatment |
| Uranium-238 | 4.5 billion yr | Geological dating |
| Polonium-210 | 138 days | Research, smoke detectors |
| Radon-222 | 3.8 days | Indoor air quality testing |
Unit 6 Key Vocabulary
Base (b)
The constant factor in f(x) = abˣ; must satisfy b > 0, b ≠ 1.
Initial Value (a)
The y-intercept of f(x) = abˣ; equals f(0).
Exponential Growth
f(x) = a(1+r)ˣ where b > 1; output increases as x increases.
Exponential Decay
f(x) = a(1−r)ˣ where 0 < b < 1; output decreases as x increases.
Growth Factor
b = 1 + r; the multiplier applied each period in a growth model.
Decay Factor
b = 1 − r; a number between 0 and 1 in a decay model.
Horizontal Asymptote
The line y = 0 that exponential graphs approach but never cross.
Scientific Notation
A number written as a × 10ⁿ where 1 ≤ |a| < 10.
Rational Exponent
An exponent of the form m/n; equivalent to ⁿ√aᵐ.
Radical
An expression using a root symbol; ⁿ√a = a^(1/n).
Compound Interest
A = P(1 + r/n)^(nt); interest earned on principal and prior interest.
Half-Life
Time for a quantity to decrease to half its original value.
Product Rule
aⁿ · aᵐ = aⁿ⁺ᵐ; add exponents when multiplying same base.
Quotient Rule
aⁿ ÷ aᵐ = aⁿ⁻ᵐ; subtract exponents when dividing same base.
Power Rule
(aⁿ)ᵐ = aⁿᵐ; multiply exponents when raising a power to a power.
Negative Exponent
a⁻ⁿ = 1/aⁿ; moves the factor to the denominator.
Mixed Review Problems
Mixed Review — Problems 1–10
| # | Topic | Problem | Answer |
|---|---|---|---|
| 1 | Exponent Rules | Simplify: x⁵ · x⁻² | x³ |
| 2 | Exponent Rules | Simplify: (3x²y)³ | 27x⁶y³ |
| 3 | Exponent Rules | Simplify: (a⁴b⁻²) / (a²b³) | a²/b⁵ |
| 4 | Scientific Notation | Write 0.00045 in scientific notation | 4.5 × 10⁻⁴ |
| 5 | Scientific Notation | (3 × 10⁴)(2 × 10³) = ? | 6 × 10⁷ |
| 6 | Rational Exponents | Evaluate: 27^(2/3) | 9 |
| 7 | Rational Exponents | Write ⁴√x³ using rational exponents | x^(3/4) |
| 8 | Exponential Fns | Evaluate f(3) for f(x) = 2·3ˣ | 54 |
| 9 | Exponential Fns | Is f(x) = 5·(0.8)ˣ growth or decay? | Decay (b = 0.8 < 1) |
| 10 | Growth & Decay | Population 500 grows 4%/yr. After 3 yr? | ≈ 562.4 |
Mixed Review — Problems 11–20
| # | Topic | Problem | Answer |
|---|---|---|---|
| 11 | Exponent Rules | Simplify: (2x³)⁴ / (4x⁵) | 4x⁷ |
| 12 | Scientific Notation | (8 × 10⁶) ÷ (4 × 10²) = ? | 2 × 10⁴ |
| 13 | Rational Exponents | Simplify: (16x⁴)^(3/4) | 8x³ |
| 14 | Exponential Fns | Write equation: passes (0,3), (1,12) | f(x) = 3·4ˣ |
| 15 | Growth & Decay | Car worth $20,000 depreciates 15%/yr. After 4 yr? | ≈ $10,440 |
| 16 | Exponent Rules | Simplify: (x⁻³y²)⁻² | x⁶/y⁴ |
| 17 | Scientific Notation | Add: 3.2 × 10⁵ + 1.8 × 10⁵ | 5.0 × 10⁵ |
| 18 | Rational Exponents | Evaluate: (−8)^(1/3) | −2 |
| 19 | Compound Interest | $2,000 at 6% compounded monthly, 5 yr | ≈ $2,697.70 |
| 20 | Half-Life | 400g, half-life 3 yr. After 9 yr? | 50 g |
Worked Examples
Simplify completely: (3x²y⁻¹)³ · (x⁻²y²)²
Step 1: (3x²y⁻¹)³ = 27x⁶y⁻³
Step 2: (x⁻²y²)² = x⁻⁴y⁴
Step 3: 27x⁶y⁻³ · x⁻⁴y⁴ = 27x²y¹
A town of 5,000 grows at 3% per year. Write the model and find the population after 10 years.
Model: f(t) = 5000(1.03)^t
f(10) = 5000(1.03)^10
(1.03)^10 ≈ 1.3439
f(10) ≈ 5000 · 1.3439 ≈ 6,720
$4,000 invested at 5% compounded quarterly. Find the balance after 6 years.
A = P(1 + r/n)^(nt)
A = 4000(1 + 0.05/4)^(4·6)
A = 4000(1.0125)^24
(1.0125)^24 ≈ 1.3474
A ≈ 4000 · 1.3474 ≈ $5,390
Common Mistakes
Applying exponent rules to different bases — e.g., x³ · y² = (xy)⁵.
Product and quotient rules only apply when the bases are identical.
Using (1 + r) for exponential decay — e.g., A(t) = 500(1.12)^t for 12% decay.
Decay uses (1 − r): A(t) = 500(0.88)^t. Growth uses (1 + r).
Confusing x^(1/2) with x/2.
x^(1/2) = √x. The denominator of a rational exponent is the root index.
Adding scientific notation numbers without first matching the powers of 10.
Rewrite both numbers with the same exponent before adding or subtracting the coefficients.
Guided Practice
Guided Practice Video: Unit 6 Review
Watch the guided practice walkthrough for the Unit 6 review, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Simplify: (2a³b⁻²)⁴ / (4a²b)
Hint: Apply the power rule to the numerator first, then use the quotient rule.
Write the exponential equation for a population that starts at 1,200 and decreases by 8% each year.
Hint: Use the decay model f(t) = a(1 − r)^t. Identify a and r first.
Evaluate: (125)^(2/3) + (16)^(3/4)
Hint: Convert each to radical form: ³√125² and ⁴√16³. Simplify the roots first.
Practice Problems
Simplify: (x⁴y⁻³)² · (x⁻¹y²)³
Write 0.0000000056 in scientific notation.
Evaluate: 32^(3/5)
Write the equation for a function with initial value 7 that grows by factor 2 each step.
A sample of 1,000g has a half-life of 4 years. How much remains after 12 years?
$6,000 at 3.5% compounded monthly for 5 years. Find the balance.
Simplify: (27x⁶)^(2/3)
Identify: is f(x) = 4·(1.15)ˣ growth or decay? State the rate.
(6 × 10⁻³)(4 × 10⁵) — express in scientific notation.
Find the y-intercept and asymptote of f(x) = 9·(0.6)ˣ.
Real-World Applications
Bacteria Colony
Setup: A lab starts with 200 bacteria. The colony doubles every 4 hours.
Question: How many bacteria are there after 24 hours?
Model: f(t) = 200 · 2^(t/4)
Solution: f(24) = 200 · 2^6 = 200 · 64 = 12,800 bacteria
Car Depreciation
Setup: A car costs $35,000 and loses 18% of its value each year.
Question: What is the car worth after 5 years?
Model: f(t) = 35,000 · (0.82)^t
Solution: f(5) = 35,000 · (0.82)⁵ ≈ 35,000 · 0.3707 ≈ $12,975
Savings Account
Setup: $5,000 invested at 4.5% compounded quarterly.
Question: What is the balance after 10 years?
Model: A = 5000(1 + 0.045/4)^(4·10)
Solution: A = 5000(1.01125)^40 ≈ 5000 · 1.5666 ≈ $7,833
Radioactive Decay
Setup: A 600g sample has a half-life of 5 years.
Question: How much remains after 20 years?
Model: f(t) = 600 · (0.5)^(t/5)
Solution: f(20) = 600 · (0.5)^4 = 600 · 0.0625 = 37.5 g
Viral Video
Setup: A video gets 500 views on day 1 and triples each day.
Question: How many views on day 7?
Model: f(d) = 500 · 3^(d−1)
Solution: f(7) = 500 · 3⁶ = 500 · 729 = 364,500 views
Error Analysis
Error Analysis — Common Mistakes
Problem: Simplify: x³ · x⁴
✗ x¹² (multiplied exponents)
✓ x⁷ (added exponents)
Rule: Product Rule: add exponents when multiplying same base.
Problem: Write 0.0052 in scientific notation
✗ 52 × 10⁻⁴ (coefficient not between 1 and 10)
✓ 5.2 × 10⁻³
Rule: Coefficient must satisfy 1 ≤ |a| < 10.
Problem: Evaluate 8^(2/3)
✗ 8^(2/3) = 8² / 3 = 64/3 (divided by 3)
✓ 8^(2/3) = (³√8)² = 2² = 4
Rule: Denominator = index of radical; numerator = power.
Problem: Is f(x) = 3·(−2)ˣ exponential?
✗ Yes — it has the form abˣ
✓ No — base must be positive (b > 0)
Rule: Exponential functions require b > 0 and b ≠ 1.
Challenge Problems
Challenge 1
Simplify completely: (2x²y⁻³)⁴ · (x⁻¹y²)⁻²
(2x²y⁻³)⁴ = 16x⁸y⁻¹²
(x⁻¹y²)⁻² = x²y⁻⁴
16x⁸y⁻¹² · x²y⁻⁴ = 16x¹⁰y⁻¹⁶
Answer: 16x¹⁰ / y¹⁶
Challenge 2
(6.4 × 10⁸) ÷ (3.2 × 10⁻³) — express in scientific notation
6.4 ÷ 3.2 = 2
10⁸ ÷ 10⁻³ = 10^(8−(−3)) = 10¹¹
Answer: 2 × 10¹¹
Challenge 3
Solve: 4^x = 8 (write x as a fraction)
4^x = 8 → (2²)^x = 2³ → 2^(2x) = 2³
2x = 3
Answer: x = 3/2
Challenge 4
A population doubles every 12 years. How long to triple from 1,000?
1000 · 2^(t/12) = 3000
2^(t/12) = 3
t/12 = log₂(3) ≈ 1.585
t ≈ 19.02 years
Answer: ≈ 19 years
Challenge 5
Simplify: (27^(2/3) · 8^(1/3)) / (9^(1/2) · 4^(3/2))
27^(2/3) = 9, 8^(1/3) = 2, 9^(1/2) = 3, 4^(3/2) = 8
(9 · 2) / (3 · 8) = 18/24
Answer: 3/4