Exponents and Powers

What Are Exponents?

An exponent (also called a power or index) is a mathematical shorthand that tells you how many times to multiply a number by itself. Instead of writing 2 × 2 × 2 × 2, we can write 2⁴, which is much more efficient.

Anatomy of an Exponential Expression

In the expression 2⁵:

• 2 is the base (the number being multiplied)
• 5 is the exponent (how many times to multiply the base by itself)
• 32 is the power or value (the result)

We read 2⁵ as:

• “Two to the power of five”
• “Two to the fifth power”
• “Two raised to the fifth”

Understanding the Concept

2⁵ = 2 × 2 × 2 × 2 × 2 = 32

The exponent tells us: “Use 2 as a factor 5 times.”

Think of exponents as a way to express repeated multiplication, just as multiplication is a way to express repeated addition:

• Addition: 2 + 2 + 2 + 2 = 4 × 2 = 8
• Multiplication: 2 × 2 × 2 × 2 = 2⁴ = 16

Special Exponent Names

Squared (Exponent of 2)

When a number has an exponent of 2, we say it’s “squared.” This name comes from geometry: a square with side length 5 has an area of 5² = 25 square units.

Examples:

• 3² = 3 × 3 = 9 (“three squared”)
• 10² = 10 × 10 = 100
• 12² = 12 × 12 = 144

Cubed (Exponent of 3)

When a number has an exponent of 3, we say it’s “cubed.” This name comes from geometry: a cube with edge length 4 has a volume of 4³ = 64 cubic units.

Examples:

• 2³ = 2 × 2 × 2 = 8 (“two cubed”)
• 5³ = 5 × 5 × 5 = 125
• 10³ = 10 × 10 × 10 = 1,000

Powers of Ten

Powers of 10 are especially important because our number system is base 10:

• 10¹ = 10
• 10² = 100
• 10³ = 1,000
• 10⁴ = 10,000
• 10⁵ = 100,000
• 10⁶ = 1,000,000

Pattern: The exponent tells you how many zeros follow the 1!

Special Exponent Rules

The Power of 1

Any number raised to the power of 1 equals itself.

Examples:

• 5¹ = 5
• 100¹ = 100
• 999¹ = 999

Why? The exponent 1 means “use the base as a factor once,” which is just the number itself.

The Power of 0

Any number (except 0) raised to the power of 0 equals 1.

Examples:

• 5⁰ = 1
• 100⁰ = 1
• 999⁰ = 1

Why? This follows from the division law of exponents. For example: 5³ ÷ 5³ = 1 (anything divided by itself is 1), but also 5³ ÷ 5³ = 5³⁻³ = 5⁰, so 5⁰ must equal 1.

Base of 1

1 raised to any power equals 1.

Examples:

• 1² = 1 × 1 = 1
• 1⁵ = 1 × 1 × 1 × 1 × 1 = 1
• 1¹⁰⁰ = 1

Why? Multiplying 1 by itself any number of times always gives 1.

Base of 0

0 raised to any positive power equals 0.

Examples:

• 0² = 0 × 0 = 0
• 0⁵ = 0 × 0 × 0 × 0 × 0 = 0

Note: 0⁰ is undefined in mathematics (we don’t assign it a value).

Calculating Powers: Worked Examples

Example 1: Small Base, Small Exponent

Calculate: 3⁴

Solution: 81

Explanation:

3⁴ = 3 × 3 × 3 × 3
   = 9 × 3 × 3
   = 27 × 3
   = 81

Example 2: Base of 10

Calculate: 10⁵

Solution: 100,000

Explanation: 10⁵ = 10 × 10 × 10 × 10 × 10 = 100,000 Quick method: The exponent is 5, so write 1 followed by 5 zeros.

Example 3: Larger Base

Calculate: 5⁴

Solution: 625

Explanation:

5⁴ = 5 × 5 × 5 × 5
   = 25 × 5 × 5
   = 125 × 5
   = 625

Example 4: Negative Base with Even Exponent

Calculate: (-2)⁴

Solution: 16

Explanation:

(-2)⁴ = (-2) × (-2) × (-2) × (-2)
      = 4 × (-2) × (-2)
      = (-8) × (-2)
      = 16

Key insight: A negative number raised to an even power is positive.

Example 5: Negative Base with Odd Exponent

Calculate: (-2)³

Solution: -8

Explanation:

(-2)³ = (-2) × (-2) × (-2)
      = 4 × (-2)
      = -8

Key insight: A negative number raised to an odd power is negative.

Index Laws (Laws of Exponents)

Index laws are rules that make working with exponents much easier. They’re essential for algebra and higher mathematics.

Law 1: Multiplying Powers with the Same Base

When multiplying powers with the same base, add the exponents.

Formula: aᵐ × aⁿ = aᵐ⁺ⁿ

Example 6: Multiplying Same Bases

Simplify: 2³ × 2⁴

Solution: 2⁷ = 128

Explanation:

Method 1 (Using the law):
2³ × 2⁴ = 2³⁺⁴ = 2⁷ = 128

Method 2 (Expanding to verify):
2³ × 2⁴ = (2×2×2) × (2×2×2×2)
        = 2×2×2×2×2×2×2
        = 2⁷
        = 128

Example 7: Three Terms

Simplify: 5² × 5³ × 5¹

Solution: 5⁶ = 15,625

Explanation: 5² × 5³ × 5¹ = 5²⁺³⁺¹ = 5⁶ = 15,625

Law 2: Dividing Powers with the Same Base

When dividing powers with the same base, subtract the exponents.

Formula: aᵐ ÷ aⁿ = aᵐ⁻ⁿ

Example 8: Dividing Same Bases

Simplify: 3⁷ ÷ 3⁴

Solution: 3³ = 27

Explanation:

Method 1 (Using the law):
3⁷ ÷ 3⁴ = 3⁷⁻⁴ = 3³ = 27

Method 2 (Expanding to verify):
3⁷ ÷ 3⁴ = (3×3×3×3×3×3×3) ÷ (3×3×3×3)
        = 3×3×3 (after canceling four 3s)
        = 3³
        = 27

Example 9: Subtracting to Get Zero

Simplify: 7⁵ ÷ 7⁵

Solution: 7⁰ = 1

Explanation: 7⁵ ÷ 7⁵ = 7⁵⁻⁵ = 7⁰ = 1 This confirms why any number to the power of 0 equals 1!

Law 3: Power of a Power

When raising a power to another power, multiply the exponents.

Formula: (aᵐ)ⁿ = aᵐˣⁿ

Example 10: Power of a Power

Simplify: (2³)⁴

Solution: 2¹² = 4,096

Explanation:

Method 1 (Using the law):
(2³)⁴ = 2³ˣ⁴ = 2¹² = 4,096

Method 2 (Expanding to verify):
(2³)⁴ = (2³) × (2³) × (2³) × (2³)
      = 2³⁺³⁺³⁺³    (using multiplication law)
      = 2¹²
      = 4,096

Law 4: Power of a Product

When raising a product to a power, raise each factor to that power.

Formula: (ab)ⁿ = aⁿ × bⁿ

Example 11: Power of a Product

Simplify: (2 × 5)³

Solution: 1,000

Explanation:

Method 1 (Using the law):
(2 × 5)³ = 2³ × 5³ = 8 × 125 = 1,000

Method 2 (Calculating inside first):
(2 × 5)³ = (10)³ = 1,000

Law 5: Power of a Quotient

When raising a quotient to a power, raise both numerator and denominator to that power.

Formula: (a/b)ⁿ = aⁿ/bⁿ

Example 12: Power of a Quotient

Simplify: (3/2)⁴

Solution: 81/16

Explanation:

(3/2)⁴ = 3⁴/2⁴ = 81/16

Comparing Expressions

It’s important to understand when expressions are equal and when they’re different.

Example 13: Different Bases vs. Different Exponents

Compare: Which is larger, 2⁸ or 8²?

Solution: 2⁸ is larger

Explanation:

2⁸ = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256
8² = 8 × 8 = 64
Therefore, 2⁸ > 8²

Insight: A smaller base with a larger exponent can exceed a larger base with a smaller exponent!

Common Errors and How to Avoid Them

Practice Problems

Basic Calculations

1. Calculate: 2⁶ Answer: 64

2. Calculate: 5³ Answer: 125

3. Calculate: 10⁴ Answer: 10,000

4. Calculate: 3⁵ Answer: 243

5. What is 4⁰? Answer: 1

6. Calculate: 1¹⁰⁰ Answer: 1

Negative Bases

7. Calculate: (-3)³ Answer: -27

8. Calculate: (-2)⁴ Answer: 16

9. Calculate: (-5)² Answer: 25

Index Laws - Multiplication

10. Simplify: 3² × 3⁴ Answer: 3⁶ = 729

11. Simplify: 5³ × 5² × 5 Answer: 5⁶ = 15,625

12. Simplify: 2⁴ × 2³ Answer: 2⁷ = 128

Index Laws - Division

13. Simplify: 7⁸ ÷ 7⁵ Answer: 7³ = 343

14. Simplify: 4⁶ ÷ 4² Answer: 4⁴ = 256

15. Simplify: 10⁵ ÷ 10³ Answer: 10² = 100

Index Laws - Power of a Power

16. Simplify: (2²)³ Answer: 2⁶ = 64

17. Simplify: (3³)² Answer: 3⁶ = 729

18. Simplify: (5²)⁴ Answer: 5⁸ = 390,625

Mixed Problems

19. Which is greater: 3⁴ or 4³? Answer: 3⁴ = 81 is greater than 4³ = 64

20. Calculate: 2³ × 2² ÷ 2⁴ Answer: 2¹ = 2

Real-World Applications

Application 1: Computer Storage

Problem: Computer memory is measured in powers of 2. If 1 kilobyte = 2¹⁰ bytes and 1 megabyte = 2²⁰ bytes, how many kilobytes are in 1 megabyte?

Solution:

1 MB ÷ 1 KB = 2²⁰ ÷ 2¹⁰ = 2²⁰⁻¹⁰ = 2¹⁰ = 1,024 kilobytes

There are 1,024 kilobytes in 1 megabyte.

Application 2: Bacterial Growth

Problem: A bacteria population doubles every hour. If you start with 3 bacteria, how many will you have after 8 hours?

Solution:

After 1 hour: 3 × 2¹ = 6
After 2 hours: 3 × 2² = 12
After 8 hours: 3 × 2⁸ = 3 × 256 = 768 bacteria

Application 3: Paper Folding

Problem: If you fold a piece of paper in half, you have 2 layers. If you fold it again, you have 4 layers. How many layers do you have after 6 folds?

Solution:

After n folds: 2ⁿ layers
After 6 folds: 2⁶ = 64 layers

Application 4: Compound Interest

Problem: If you invest $1,000 and it grows by 10% each year, the formula is: Amount = 1000 × (1.1)ⁿ where n is the number of years. What do you have after 3 years?

Solution:

Amount = 1000 × (1.1)³
       = 1000 × 1.331
       = $1,331

Application 5: Chess Legend

According to legend, the inventor of chess asked the king for rice: 1 grain on the first square of a chessboard, 2 on the second, 4 on the third, doubling each time for all 64 squares.

Problem: How many grains on the 10th square?

Solution:

Square 1: 2⁰ = 1 grain
Square 2: 2¹ = 2 grains
Square 3: 2² = 4 grains
Square 10: 2⁹ = 512 grains

(By the 64th square, the total would be more grains than exist on Earth!)

Why This Matters

Exponents are fundamental to:

• Science: Measuring tiny atoms (10⁻¹⁰ m) and vast distances (10²⁶ m)
• Technology: Computer processing power doubles exponentially (Moore’s Law)
• Finance: Compound interest grows exponentially over time
• Biology: Population growth and bacterial reproduction
• Physics: Energy, light intensity, and sound all involve exponential relationships
• Engineering: Signal strength, structural load calculations

From understanding how viruses spread to calculating investment returns, from measuring earthquakes to exploring space, exponents are everywhere in the modern world.

Connection to Other Topics

Exponents connect to many mathematical concepts:

• Square Roots: The inverse operation of squaring (√25 = 5 because 5² = 25)
• Scientific Notation: Writing very large or small numbers (6.02 × 10²³)
• Algebra: Polynomial expressions use exponents (x² + 3x + 5)
• Geometry: Area (units²) and volume (units³) formulas
• Logarithms: The inverse operation of exponentiation (log₂ 8 = 3 because 2³ = 8)
• Sequences: Geometric sequences multiply by a constant factor each term

Tips for Success

1. Master the basics: Know your squares (1² through 15²) and cubes (1³ through 10³) by heart
2. Practice index laws: These rules are essential for algebra and beyond
3. Check your work: Calculate a different way to verify (expand or use a calculator)
4. Understand before memorizing: Know why the rules work, not just what they are
5. Watch for negative bases: Use parentheses to avoid sign errors
6. Start small: Practice with small numbers before tackling large exponents
7. Use patterns: Look for patterns in powers of 2, 3, 5, and 10

Interesting Facts

• Powers of 2: Many things in technology use powers of 2 (8, 16, 32, 64, 128, 256, 512, 1024…)
• Google’s name: Comes from “googol,” which is 10¹⁰⁰ (1 followed by 100 zeros)
• Folding paper: It’s physically impossible to fold a regular piece of paper more than 7-8 times, even though 2⁸ = 256 layers doesn’t seem that thick
• Exponential growth: Humans struggle to understand exponential growth intuitively—we think linearly

Mastering exponents opens the door to advanced mathematics, science, and understanding exponential phenomena in the world around you!