Power Calculation

What is Exponentiation?

Exponentiation is a mathematical notation that represents multiplying the same number multiple times.
Let a be the base and n be the exponent.

an

It is read as "a to the power of n."

Basic Concept

• a1 = a (just once, so unchanged)
• a2 = a × a (multiplied 2 times → square)
• a3 = a × a × a (multiplied 3 times → cube)
• a4 = a × a × a × a (multiplied 4 times → 4th power)

In other words, the exponent tells you how many times to multiply the number.

Special Cases

• a0 = 1 (when a ≠ 0)
• a-n = 1 / an
  A negative exponent means the reciprocal. Example: 2-3 = 1 / 23 = 1/8 = 0.125
• a1/2 means "square root" → Example: 91/2 = √9 = 3
• a1/3 means "cube root" → Example: 81/3 = ³√8 = 2

Everyday Examples

• Area: A square with side length 5 cm → Area = 52 = 25 cm²
• Volume: A cube with side length 3 cm → Volume = 33 = 27 cm³
• Money: If you double 100 yen, and then double it again → 100 × 22 = 400 yen

Summary

• Exponentiation = a way to express repeated multiplication
• Positive integer exponent → repeated multiplication
• Exponent 0 → 1
• Negative exponent → reciprocal
• Fractional exponent → root (square root, cube root, etc.)

Simply put, exponentiation is "a convenient way to write down how many times a number is multiplied."

1. Basic Concept

Exponentiation means multiplying the same number repeatedly.
Let a be the base and n be the exponent.

an = a × a × … × a (n times)

• 23 = 2 × 2 × 2 = 8
• 54 = 5 × 5 × 5 × 5 = 625

2. Negative Exponents

a-n = 1 / an

• 2-3 = 1 / 23 = 1/8 = 0.125
• 10-2 = 1 / 102 = 1/100 = 0.01

3. Zero Exponent

a0 = 1 (a ≠ 0)

• 50 = 1
• 1000 = 1

However, 00 is undefined.

4. Fractional Exponents (Roots)

A fractional exponent means a root.

• a1/2 = √a
• a1/3 = ³√a

Examples:

• 91/2 = √9 = 3
• 271/3 = ³√27 = 3
• 163/4 = (163)1/4 = √[4]{4096} = 8

5. Real Number Exponents (Using Logarithms)

Decimal or real number exponents are calculated using logarithms and exponential functions.
ab = eb × ln(a) (a > 0)

• 20.5 = e0.5 × ln(2) ≈ 1.414
• 101.2 = e1.2 × ln(10) ≈ 15.849

6. Efficient Calculation Method (Computer Calculation)

For large exponents, the "Exponentiation by Squaring" method is used.

Concept:

• Break the exponent into binary form and calculate efficiently
• The number of multiplications grows logarithmically with the size of the exponent, making it fast

Example: 213

• 13 = 1101 (binary)
• 213 = 28 × 24 × 21

Summary

• Positive integer exponents → repeated multiplication
• Negative exponents → reciprocal
• Zero exponent → 1 (but 0⁰ is undefined)
• Fractional exponents → roots
• Decimal/real exponents → calculated using logarithms
• Large exponents → efficient calculation using exponentiation by squaring

In short, exponentiation is "a rule for expressing repeated multiplication," with slightly different meanings depending on the type of exponent.