Page 2: Extracting Factors and Manipulating Cube Roots

The second page focuses on techniques for simplifying cube root expressions by extracting factors and manipulating cube root terms.

Vocabulary: Wyciąganie czynnika przed pierwiastek refers to the process of extracting a factor from under the cube root sign.

The page demonstrates the step-by-step process of extracting factors from ³√250:

1. Factor 250 into its prime components: 2 × 5³
2. Identify the largest perfect cube factor: 5³ = 125
3. Extract the cube root of this factor: ³√125 = 5
4. Leave the remaining factor under the cube root: ³√2

Thus, ³√250 simplifies to 5 · ³√2.

Example: ³√32 can be simplified to 2 · ³√4 because 32 = 2³ × 4

The page also covers addition and subtraction of cube root expressions:

• Only like terms (terms with the same cube root) can be added or subtracted
• 2³√2 + 4³√2 = 6³√2
• 3³√2 + 2³√4 + 2³√2 = 5³√2 + 2³√4

Highlight: Mastering these techniques allows for efficient simplification of complex cube root expressions and is essential for solving advanced algebraic problems involving pierwiastki trzeciego stopnia (roots of the third degree).

Page 1: Cube Root Fundamentals and Formulas

The first page introduces the concept of pierwiastek sześcienny (cube root) and presents essential formulas for working with cube roots.

Definition: The cube root of a number a, denoted as ³√a, is a value that, when cubed, equals a.

Key formulas presented include:

1. ³√(a² · b) = ³√a² · ³√b
2. ³√(a · b²) = ³√a · ³√b²
3. ³√a = b when b³ = a
4. (³√a)² = a²/³
5. (³√a)³ = a

Example: ³√64 = 4 because 4³ = 64

The page also provides examples of jak obliczyć pierwiastek sześcienny (how to calculate cube root):

• ³√64 = 4
• ³√$-8$ = -2
• ³√27 = 3

Highlight: Understanding these formulas is crucial for simplifying complex expressions involving cube roots and solving equations of the third degree.