Before You Begin
Essential Vocabulary
a
x
＝
m
n
a
n
a
= coefficient
x
= variable / base / radicand
m
= Power / Exponent
n
= Index
= Radical sign
General Form
Example
Base
The number or variable being multiplied repeatedly. In x⁵, the base is x.
Exponent / Power
Tells how many times the base is used as a factor. In x⁵, the exponent is 5.
Coefficient
The numerical factor in a monomial. In 5x³, the coefficient is 5.
Monomial
A single-term expression: a number, variable, or product of both (e.g., 3x²y).
Ratio of Monomials
A fraction where both numerator and denominator are monomials.
Integer Exponents
Exponents that are whole numbers, including zero and negative integers.
Core Content
The Seven Laws of Exponents
SOL Standard A.EO.3: You are expected to derive these laws through pattern exploration, not just memorize them. Notice why each rule works by thinking about repeated multiplication.
① Product Rule
xᵃ · xᵇ = x^(a+b)
Same base? Add the exponents. The base stays the same.
Worked Example
x³ · x⁵same base x
= x^(3+5)add exponents
= x⁸
② Quotient Rule
xᵃ ÷ xᵇ = x^(a−b)
Same base? Subtract the exponents. (x ≠ 0)
Worked Example
y⁷ ÷ y²same base y
= y^(7−2)subtract exponents
= y⁵
③ Power of a Power
(xᵃ)ᵇ = x^(a·b)
A power raised to a power? Multiply the exponents.
Worked Example
(m⁴)³
= m^(4·3)multiply exponents
= m¹²
④ Power of a Product
(xy)ᵃ = xᵃ · yᵃ
Distribute the exponent to every factor inside.
Worked Example
(2ab)³
= 2³ · a³ · b³distribute
= 8a³b³
⑤ Power of a Quotient
(x/y)ᵃ = xᵃ/yᵃ
Apply the exponent to both numerator and denominator. (y ≠ 0)
Worked Example
(3/n)⁴
= 3⁴ / n⁴distribute
= 81 / n⁴
⑥ Zero Exponent
x⁰ = 1
Any non-zero base to the zero power always equals 1. (x ≠ 0)
Worked Example
7⁰ = 1
(−5)⁰ = 1
(xy)⁰ = 1
⑦ Negative Exponent
x⁻ᵃ = 1 / xᵃ
A negative exponent means take the reciprocal. It does NOT make the result negative!
Worked Example
x⁻³= 1/x³
2⁻⁴= 1/2⁴ = 1/16
3a⁻²= 3/a²
Application
Simplifying Multivariable Expressions
When simplifying ratios of monomials, handle coefficients and each variable separately.
Strategy: Work in Three Steps
Step 1Simplify the coefficient
Step 2Apply quotient rule to each variable
Step 3Rewrite negative exponents
Full Example
12x⁵y³ / 4x²y⁷
= (12/4)(x⁵⁻²)(y³⁻⁷)
= 3 · x³ · y⁻⁴
= 3x³ / y⁴
Watch Out!
Common Mistakes & Corrections
✗ Wrong
x² · x³ = x⁶
Multiplied exponents instead of adding
✓ Correct
x² · x³ = x⁵
Product Rule → add the exponents
✗ Wrong
(x³)² = x⁵
Added instead of multiplying
✓ Correct
(x³)² = x⁶
Power of a Power → multiply exponents
✗ Wrong
x⁻² = −x²
Negative exponent ≠ negative number!
✓ Correct
x⁻² = 1/x²
Negative exponent means reciprocal
✗ Wrong
(2x)³ = 2x³
Forgot to apply the exponent to 2
✓ Correct
(2x)³ = 8x³
Power of Product: distribute to all factors
Quick Reference
Laws at a Glance
Exponent Laws — Cheat Sheet
Product
xᵃ · xᵇ = x^(a+b)
Quotient
xᵃ ÷ xᵇ = x^(a−b)
Power of Power
(xᵃ)ᵇ = x^(ab)
Power of Product
(xy)ᵃ = xᵃyᵃ
Power of Quotient
(x/y)ᵃ = xᵃ/yᵃ
Zero Exponent
x⁰ = 1
Negative Exponent
x⁻ᵃ = 1/xᵃ
Practice
Practice Problems with Desmos / Amplify Strategies
🖩
Desmos / Amplify Strategy
To verify any exponent expression, type it directly into the Desmos expression bar. Desmos evaluates and simplifies automatically. Use y = to graph and compare two expressions — if they produce the same graph, they are equivalent.
To verify any exponent expression, type it directly into the Desmos expression bar. Desmos evaluates and simplifies automatically. Use y = to graph and compare two expressions — if they produce the same graph, they are equivalent.
Problem
🖩 Desmos Strategy
📋 SOL Exam Desmos Tips
✦ Use ^ for exponents:
x^3✦ Negative exponents:
x^{-2}✦ Graph two expressions to check equivalence
✦ Use sliders for multi-variable expressions
✦ Type expressions exactly — parentheses matter
✦ Zoom out if graphs look flat near origin
Why It Matters
Real-World Connections
Biology — Bacteria Growth
Bacteria double every hour: 2¹ · 2³ = 2⁴ cells. The Product Rule models population growth.
Technology — Data Storage
A kilobyte is 2¹⁰ bytes; a megabyte 2²⁰. Dividing storage uses the Quotient Rule.
Science — Very Large Numbers
Scientific notation relies on power rules to multiply and divide numbers like 10⁶ × 10³ = 10⁹.
Finance — Compound Interest
A = P(1+r)ᵗ. Raising a quantity to a power uses Power of a Product rules.