Page 1: Potęga o Wykładniku Rzeczywistym

This page introduces the concept of potęga o wykładniku rzeczywistym (power with a real exponent) and provides various examples and calculations. The page is filled with mathematical expressions and equations demonstrating the use of real exponents in different scenarios.

Definition: Potęga o wykładniku rzeczywistym refers to a mathematical operation where a base number is raised to a power that can be any real number, including integers, fractions, and irrational numbers.

The page showcases several examples of calculations involving real exponents:

Example: 2^3 = 8, illustrating a simple case of an integer exponent.

Example: $2^3$^2 = 2^6 = 64, demonstrating the rule of exponents where powers are multiplied when a power is raised to another power.

Example: x^(3-2) = x^1 = x, showing how exponents can be subtracted when dividing powers with the same base.

The page also includes more complex expressions:

Example: (a·b)^x = a^x · b^x, illustrating the distributive property of exponents over multiplication.

Highlight: The page emphasizes the importance of understanding the rules of exponents, such as the product rule, quotient rule, and power rule, when dealing with real exponents.

Several equations on the page involve variables and mixed operations:

Example: $2^3$^(1/2) = 2^(3/2) ≈ 2.83, demonstrating how to handle fractional exponents.

Vocabulary: "Wykładnik rzeczywisty" (real exponent) refers to any exponent that is a real number, including rational and irrational numbers.

The page concludes with more advanced examples, including expressions with multiple variables and complex exponent manipulations, reinforcing the importance of mastering potęga o wykładniku rzeczywistym for solving sophisticated mathematical problems.