Integer Exponents

Exponents are a powerful tool in mathematics! When we expand beyond positive whole number exponents to include negative and zero exponents, we unlock even more mathematical possibilities. Let's explore how integer exponents work!

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What Are Integer Exponents?

Integer exponents include:

• Positive integers: 2³, 5⁴, 10²
• Zero: 5⁰, 10⁰, x⁰
• Negative integers: 2⁻³, 5⁻², 10⁻¹

All follow consistent rules that make calculations easier!

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Review: Positive Exponents

Base = the number being multiplied Exponent = how many times to multiply it

Example: 2⁴ = 2 × 2 × 2 × 2 = 16

General form: aⁿ where a is the base, n is the exponent

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Zero as an Exponent

Rule: Any non-zero number raised to the power of 0 equals 1.

a⁰ = 1 (where a ≠ 0)

Examples:

• 5⁰ = 1
• 10⁰ = 1
• 237⁰ = 1
• (-8)⁰ = 1
• x⁰ = 1

Why? Using the division property of exponents:

• 5³ ÷ 5³ = 125 ÷ 125 = 1
• 5³ ÷ 5³ = 5³⁻³ = 5⁰
• Therefore, 5⁰ = 1

Special note: 0⁰ is undefined in mathematics.

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Negative Exponents

Rule: A negative exponent means "take the reciprocal."

a⁻ⁿ = 1/aⁿ

Examples:

Example 1: 2⁻³ 2⁻³ = 1/2³ = 1/8

Example 2: 5⁻² 5⁻² = 1/5² = 1/25

Example 3: 10⁻¹ 10⁻¹ = 1/10¹ = 1/10 = 0.1

Example 4: 3⁻⁴ 3⁻⁴ = 1/3⁴ = 1/81

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Negative Exponents with Fractions

Rule: (a/b)⁻ⁿ = (b/a)ⁿ

Flip the fraction and make the exponent positive!

Examples:

Example 1: (2/3)⁻² (2/3)⁻² = (3/2)² = 9/4

Example 2: (1/4)⁻³ (1/4)⁻³ = (4/1)³ = 4³ = 64

Example 3: (5/2)⁻¹ (5/2)⁻¹ = (2/5)¹ = 2/5

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Negative Exponents in Fractions

If a negative exponent appears in the numerator or denominator:

Rule 1: 1/a⁻ⁿ = aⁿ Move from denominator to numerator and change sign!

Example: 1/2⁻³ = 2³ = 8

Rule 2: a⁻ⁿ/b = 1/(aⁿb)

Example: 3⁻²/5 = 1/(3²·5) = 1/(9·5) = 1/45

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Laws of Exponents with Integers

All exponent rules work with integer exponents!

Product Rule: aᵐ · aⁿ = aᵐ⁺ⁿ

Example: 2³ · 2⁻⁵ = 2³⁺⁽⁻⁵⁾ = 2⁻² = 1/4

Quotient Rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ

Example: 5² ÷ 5⁵ = 5²⁻⁵ = 5⁻³ = 1/125

Power Rule: (aᵐ)ⁿ = aᵐⁿ

Example: (3⁻²)³ = 3⁻⁶ = 1/729

Power of a Product: (ab)ⁿ = aⁿbⁿ

Example: (2x)⁻³ = 2⁻³x⁻³ = x⁻³/8

Power of a Quotient: (a/b)ⁿ = aⁿ/bⁿ

Example: (2/3)⁻² = 2⁻²/3⁻² = 3²/2² = 9/4

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Simplifying Expressions with Integer Exponents

Example 1: Simplify x⁻⁴ · x⁷

Solution: Use product rule: x⁻⁴⁺⁷ = x³

Answer: x³

Example 2: Simplify (2a⁻³b²)⁻²

Solution: Apply power rule: (2a⁻³b²)⁻² = 2⁻² · a⁶ · b⁻⁴

Convert to positive exponents: = a⁶/(2² · b⁴) = a⁶/(4b⁴)

Answer: a⁶/(4b⁴)

Example 3: Simplify (3x²y⁻¹)/(6x⁻³y⁴)

Solution: Separate coefficients and variables: = (3/6) · (x²/x⁻³) · (y⁻¹/y⁴)

Simplify each part: = (1/2) · x²⁻⁽⁻³⁾ · y⁻¹⁻⁴ = (1/2) · x⁵ · y⁻⁵ = x⁵/(2y⁵)

Answer: x⁵/(2y⁵)

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Writing with Positive Exponents

Often we want to rewrite expressions using only positive exponents.

Example 1: Write 5x⁻³ with positive exponents

Solution: 5x⁻³ = 5/x³

Example 2: Write 2a⁻²b⁴/c⁻³ with positive exponents

Solution: Move negative exponents: = 2b⁴c³/a²

Example 3: Write (m⁻²n³)⁻¹ with positive exponents

Solution: Apply power rule: = m²n⁻³ = m²/n³

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Real-World Applications

Scientific Notation: Negative exponents represent very small numbers

• 1 millimeter = 10⁻³ meters
• 1 microsecond = 10⁻⁶ seconds
• Diameter of a cell: 10⁻⁵ meters

Computer Science: Data sizes

• 1 byte = 2⁰ bytes
• 1 kilobyte = 2¹⁰ bytes
• 1 megabyte = 2²⁰ bytes

Finance: Compound interest formulas use exponents

• Amount = P(1 + r)⁻ⁿ for present value

Physics: Inverse square laws

• Light intensity ∝ distance⁻²
• Gravitational force ∝ distance⁻²

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Common Mistakes to Avoid

❌ Mistake 1: Thinking 5⁰ = 0

• Wrong: 5⁰ = 0
• Right: 5⁰ = 1

❌ Mistake 2: Making the base negative instead of reciprocal

• Wrong: 2⁻³ = -8
• Right: 2⁻³ = 1/8

❌ Mistake 3: Forgetting to flip when dividing

• Wrong: 1/3⁻² = 1/9
• Right: 1/3⁻² = 3² = 9

❌ Mistake 4: Applying exponent to coefficient incorrectly

• Wrong: 2x⁻³ = 1/(2x³) ... NO! Only x has negative exponent
• Right: 2x⁻³ = 2/x³

❌ Mistake 5: Adding exponents when multiplying different bases

• Wrong: 2³ · 3² = 6⁵
• Right: 2³ · 3² = 8 · 9 = 72

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Practice Strategy

Step 1: Identify negative and zero exponents

Step 2: Apply definitions:

• a⁰ = 1
• a⁻ⁿ = 1/aⁿ

Step 3: Use exponent laws to combine like bases

Step 4: Convert to positive exponents if required

Step 5: Simplify fully

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Quick Reference

Zero Exponent:

• a⁰ = 1

Negative Exponent:

• a⁻ⁿ = 1/aⁿ
• 1/a⁻ⁿ = aⁿ
• (a/b)⁻ⁿ = (b/a)ⁿ

Exponent Laws:

• aᵐ · aⁿ = aᵐ⁺ⁿ
• aᵐ ÷ aⁿ = aᵐ⁻ⁿ
• (aᵐ)ⁿ = aᵐⁿ
• (ab)ⁿ = aⁿbⁿ
• (a/b)ⁿ = aⁿ/bⁿ

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Summary

Integer exponents extend our understanding beyond positive whole numbers:

Zero exponents: Any non-zero base to the power of 0 equals 1

• 5⁰ = 1, x⁰ = 1

Negative exponents: Represent reciprocals

• 2⁻³ = 1/8
• x⁻² = 1/x²

All exponent laws apply to integer exponents, making calculations consistent and predictable.

Key skills:

• Convert negative exponents to positive
• Simplify expressions using exponent laws
• Recognize real-world applications

Understanding integer exponents is essential for algebra, scientific notation, and advanced mathematics!