Calculating Powers with Negative Exponents

This page focuses on solving problems involving potęga o wykładniku całkowitym ujemnym (powers with negative integer exponents). It presents a series of exercises that demonstrate various techniques for simplifying and calculating these expressions.

Definition: A power with a negative exponent is equivalent to its reciprocal with a positive exponent. For example, x^$-n$ = 1 / $x^n$.

The exercises cover a range of scenarios, including:

1. Simple negative exponents with integer bases
2. Fractional bases with negative exponents
3. Negative bases with negative exponents
4. Compound expressions involving negative exponents

Example: 2^$-3$ = $1/2$^3 = 1/8

Example: $9/11$^$-3$ = $11/9$^3 = 1331/729

Highlight: The document emphasizes the importance of converting negative exponents to positive ones by taking the reciprocal of the base and changing the sign of the exponent.

Some more complex problems are also presented, such as:

• $5/4$^$-2$ = $4/5$^2 = 16/25
• $-3$^$-3$ = -1/27
• $1/2$^$-2$ = 2^2 = 4

Vocabulary: Potęga o wykładniku wymiernym (power with a rational exponent) is also introduced, extending the concept to fractional exponents.

The exercises progressively increase in difficulty, providing a comprehensive practice set for mastering potęgi ujemne liczby (negative powers of numbers) and related concepts.