Powers with Natural Exponents

Legend has it: the inventor of chess asked the king for a reward, grains on a chessboard, doubling on each square. The king agreed, not realizing that by the 64th square there would be more grain than exists on Earth. Exponential growth is deceptive, and it governs much of our world.

• **Finance:** compound interest and investments
• **Biology:** population growth
• **Technology:** Moore's Law, computer memory

What is a Power

SHA-256 (Bitcoin, 2009) performs 64 rounds of modular exponentiation, processing 2³² hashes per second on ASIC hardware. Instead of writing 2 × 2 × 2 × 2 × 2, we write 2⁵ and read it as 'two to the fifth power'. A **power** is shorthand for repeated multiplication of a number by itself.

**aⁿ = a × a × a × ... × a** (n times) a, base n, exponent 2³ = 2 × 2 × 2 = 8 5² = 5 × 5 = 25

The names 'square' and 'cube' come from geometry: the area of a square with side a is a², the volume of a cube is a³.

Multiplying Powers

Multiplying 2³ by 2⁴: powers with the same base multiply simply, the exponents are added.

**Remember:** • Multiplication → exponents are added • Power of a power → exponents are multiplied Don't confuse them! aᵐ × aⁿ ≠ aᵐˣⁿ

Important: these rules only work with the same base. 2³ × 3⁴ cannot be simplified by adding the exponents!

Dividing Powers

If exponents are added when multiplying, they are subtracted when dividing. That makes sense: division is the inverse of multiplication.

**Common mistakes:** • (a + b)² ≠ a² + b² (it's (a+b)² = a² + 2ab + b²) • aᵐ + aⁿ ≠ aᵐ⁺ⁿ (exponents cannot be added this way!) • aᵐ × bᵐ ≠ (ab)²ᵐ (only (ab)ᵐ)

For now we only consider the case m > n in division. What happens when m = n or m < n, we'll cover that in the next lesson on integer exponents.

Exponential Growth

Powers grow incredibly fast. This is called **exponential growth**, and it's everywhere: from bacteria reproducing to compound interest.

**Where exponential growth occurs:** • Compound interest (money) • Bacterial reproduction • Virus spread • Nuclear chain reaction • Moore's Law (transistors in chips)

Exponential growth is hard to grasp intuitively. But understanding powers helps: if something doubles every year, in 10 years there will be 1024 times more!

Key Ideas

• aⁿ, a multiplied by itself n times
• aᵐ × aⁿ = aᵐ⁺ⁿ (multiplication → add exponents)
• aᵐ ÷ aⁿ = aᵐ⁻ⁿ (division → subtract exponents)
• (aᵐ)ⁿ = aᵐˣⁿ (power of a power → multiply exponents)

Related Topics

Powers are the foundation for many areas:

• Integer Exponents — Zero and negative exponents
• Roots — The inverse of powers
• Scientific Notation — Writing large numbers

Вопросы для размышления

• Why are exponents added when multiplying powers?
• What processes in life grow exponentially?
• Why is (a+b)² ≠ a² + b²?

Связанные уроки

• prob-07-expectation