Powers and Their Properties

This page provides a comprehensive overview of powers (potęgi) in mathematics, covering various types of exponents and the rules for operating with powers.

The concept of a power is introduced, defining it as a base 'a' raised to an exponent 'n', written as a^n. The page then delves into different types of exponents:

1. Natural exponents: For a ≠ 0 and n ∈ N, a^n = a · a · ... · a (n times).
2. Zero exponent: For a ≠ 0, a^0 = 1.
3. Negative integer exponents: For a ≠ 0 and n ∈ N, a^(-n) = 1 / a^n.
4. Rational exponents: For a > 0 and m, n ∈ N, a^(m/n) = ⁿ√(a^m).

Example: 2^3 = 2 · 2 · 2 = 8, demonstrating a natural exponent.

Vocabulary: Wykładnik potęgi refers to the exponent in a power expression.

The page also covers the fundamental rules for operating with powers:

1. Multiplication of powers with the same base: a^m · a^n = a^(m+n)
2. Division of powers with the same base: a^m / a^n = a^(m-n), for a ≠ 0
3. Power of a power: (a^m)^n = a^(m·n)
4. Power of a product: (a · b)^n = a^n · b^n
5. Power of a quotient: (a / b)^n = a^n / b^n, for b ≠ 0

Highlight: These zasady potęgowania are essential for simplifying complex expressions involving powers.

The page concludes by mentioning that these rules apply to all types of exponents, including natural, integer, and rational numbers, as well as real numbers for positive bases.

Definition: Potęga o wykładniku naturalnym refers to a power with a natural number as its exponent.

This comprehensive overview provides a solid foundation for understanding and working with powers in various mathematical contexts.