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Properties and operations with exponents
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An exponent (or power) tells how many times to multiply a number by itself.
Notation: bⁿ
Examples:
Positive Integer Exponents: xⁿ means multiply x by itself n times
When multiplying with the same base, add the exponents.
Rule: xᵃ · xᵇ = xᵃ⁺ᵇ
x³ · x² = (x·x·x) · (x·x) = x⁵
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Examples:
Example 1: x⁴ · x³ = x⁴⁺³ = x⁷
Example 2: 2³ · 2⁵ = 2³⁺⁵ = 2⁸ = 256
Example 3: a² · a⁷ · a = a² · a⁷ · a¹ = a²⁺⁷⁺¹ = a¹⁰
Example 4: 3x⁴ · 5x² = (3 · 5)(x⁴ · x²) = 15x⁶
Important: Bases must be the same! x³ · y² cannot be simplified using this rule
When dividing with the same base, subtract the exponents.
Rule: xᵃ ÷ xᵇ = xᵃ⁻ᵇ (when a > b)
Why? x⁵/x² = (x·x·x·x·x)/(x·x) = x³
Examples:
Example 1: x⁷ ÷ x³ = x⁷⁻³ = x⁴
Example 2: 2⁸/2⁵ = 2⁸⁻⁵ = 2³ = 8
Example 3: a¹⁰/a⁴ = a¹⁰⁻⁴ = a⁶
Example 4: 12x⁵/3x² = (12/3)(x⁵/x²) = 4x³
When raising a power to a power, multiply the exponents.
Rule: (xᵃ)ᵇ = xᵃᵇ
Why? (x²)³ = x² · x² · x² = x⁶
Examples:
Example 1: (x³)⁴ = x³ˣ⁴ = x¹²
Example 2: (a²)⁵ = a¹⁰
Example 3: (2³)² = 2⁶ = 64
Example 4: (y⁴)⁷ = y²⁸
When raising a product to a power, raise each factor to that power.
Rule: (xy)ⁿ = xⁿyⁿ
Examples:
Example 1: (xy)³ = x³y³
Example 2: (2a)⁴ = 2⁴a⁴ = 16a⁴
Example 3: (3x²)³ = 3³(x²)³ = 27x⁶
Example 4: (-2ab)³ = (-2)³a³b³ = -8a³b³
When raising a quotient to a power, raise both numerator and denominator to that power.
Rule: (x/y)ⁿ = xⁿ/yⁿ
Examples:
Example 1: (x/y)² = x²/y²
Example 2: (a/3)³ = a³/3³ = a³/27
Example 3: (2x/5)² = (2x)²/5² = 4x²/25
Example 4: (3a²/b)³ = (3a²)³/b³ = 27a⁶/b³
Any non-zero number raised to the power of zero equals 1.
Rule: x⁰ = 1 (where x ≠ 0)
Why? Using quotient rule: x³/x³ = x³⁻³ = x⁰ But x³/x³ = 1, so x⁰ = 1
Examples:
Example 1: 5⁰ = 1
Example 2: x⁰ = 1
Example 3: (3ab)⁰ = 1
Example 4: 7x⁰ = 7(1) = 7
Warning: 0⁰ is undefined!
A negative exponent means "reciprocal."
Rule: x⁻ⁿ = 1/xⁿ (where x ≠ 0)
Examples:
Example 1: x⁻³ = 1/x³
Example 2: 2⁻⁴ = 1/2⁴ = 1/16
Example 3: 5⁻² = 1/5² = 1/25
Example 4: (1/x)⁻² = x²
Rule for fractions: (a/b)⁻ⁿ = (b/a)ⁿ
Example: (2/3)⁻² = (3/2)² = 9/4
Move a factor with an exponent by changing the sign of the exponent.
Examples:
Example 1: 1/x⁻³ = x³
Example 2: 2/x⁻² = 2x²
Example 3: x⁻⁴/y⁻² = y²/x⁴
Example 4: 3x⁻²y³ = 3y³/x²
Most problems require using several rules together.
Example 1: Simplify (x²y³)⁴
= (x²)⁴(y³)⁴ [Power of product] = x⁸y¹²
Example 2: Simplify (2x³)²(3x⁴)
= 2²(x³)² · 3x⁴ [Power of product] = 4x⁶ · 3x⁴ = 12x¹⁰ [Product rule]
Example 3: Simplify (x⁵/x²)³
= (x⁵⁻²)³ [Quotient rule] = (x³)³ = x⁹ [Power rule]
Example 4: Simplify (2a³b⁻²)⁻³
= 2⁻³(a³)⁻³(b⁻²)⁻³ [Power of product] = (1/8)a⁻⁹b⁶ = b⁶/(8a⁹) [Negative exponent]
Goal: Write with positive exponents, simplified completely.
Example 1: Simplify x⁻³ · x⁷
= x⁻³⁺⁷ [Product rule] = x⁴
Example 2: Simplify (3x⁻²y⁴)/(6x³y⁻¹)
= (3/6)(x⁻²/x³)(y⁴/y⁻¹) = (1/2)x⁻⁵y⁵ = y⁵/(2x⁵)
Example 3: Simplify (a²b⁻³c⁰)²
= (a²)²(b⁻³)²(c⁰)² = a⁴b⁻⁶ · 1 = a⁴/b⁶
Example 4: Simplify 10x⁵y⁻²/(2x⁻³y⁴)
= (10/2)(x⁵/x⁻³)(y⁻²/y⁴) = 5x⁸y⁻⁶ = 5x⁸/y⁶
Scientific notation uses powers of 10: a × 10ⁿ where 1 ≤ a < 10
Examples:
Example 1: 3,400 = 3.4 × 10³
Example 2: 0.0056 = 5.6 × 10⁻³
Example 3: 7,800,000 = 7.8 × 10⁶
Operations in Scientific Notation:
Multiplication: (2 × 10³)(3 × 10⁵) = 6 × 10⁸
Division: (8 × 10⁶)/(2 × 10²) = 4 × 10⁴
Adding exponents when multiplying bases Wrong: 2³ · 3² = 6⁵ Right: 2³ · 3² = 8 · 9 = 72
Multiplying exponents with product rule Wrong: x³ · x² = x⁶ Right: x³ · x² = x⁵
Forgetting to distribute exponents Wrong: (2x)³ = 2x³ Right: (2x)³ = 8x³
Confusing negative exponent with negative number x⁻² ≠ -x² x⁻² = 1/x²
Thinking x⁰ = 0 Wrong: 5⁰ = 0 Right: 5⁰ = 1
Not simplifying negative exponents Leave answer as x⁻³ instead of 1/x³
Remember PEMDAS - exponents come before multiplication/division.
Example 1: Evaluate 2 · 3² = 2 · 9 = 18 NOT (2 · 3)² = 36
Example 2: Evaluate -2⁴ = -(2⁴) = -16 NOT (-2)⁴ = 16
Example 3: 3 + 2³ = 3 + 8 = 11
| Rule | Formula | Example |
|---|---|---|
| Product | xᵃ · xᵇ = xᵃ⁺ᵇ | x² · x³ = x⁵ |
| Quotient | xᵃ/xᵇ = xᵃ⁻ᵇ | x⁵/x² = x³ |
| Power | (xᵃ)ᵇ = xᵃᵇ | (x²)³ = x⁶ |
| Power of Product | (xy)ⁿ = xⁿyⁿ | (2x)³ = 8x³ |
| Power of Quotient | (x/y)ⁿ = xⁿ/yⁿ | (x/2)² = x²/4 |
| Zero | x⁰ = 1 | 7⁰ = 1 |
| Negative | x⁻ⁿ = 1/xⁿ | 2⁻³ = 1/8 |
Master one rule at a time Practice each rule separately first
Recognize patterns Same base? Product or quotient rule Power to power? Power rule
Work step-by-step Don't skip steps when learning
Check your work Substitute numbers to verify
Practice mixed problems Real problems use multiple rules
Write exponents clearly Use proper notation to avoid errors
Simplify completely Final answer should have positive exponents only
Is it simplified?
Use the product rule: add the exponents
Answer:
Simplify:
Step 1: Divide the coefficients
Step 2: Use the quotient rule for the variables
Simplify:
Use the power of a product rule and power rule:
Answer:
Step 3: Combine
Answer: