Subtracting Square Roots and Complex Root Operations
This page delves into the intricacies of odejmowanie pierwiastków and provides a comprehensive look at various operations involving roots. The fundamental rule presented is that roots can only be subtracted when they are of the same degree and have identical radicands.
Definition: Odejmowanie pierwiastków (subtracting roots) is only possible when the roots share the same degree and radicand.
The page then proceeds to offer a series of increasingly complex examples that demonstrate not just subtraction, but also dodawanie pierwiastków (addition of roots) and combinations of these operations.
Example: 2√6 + √6 = 3√6
This simple example illustrates the addition of roots with the same radicand.
Example: √16² - √√2² = 2√√2 - ²³√2 = ³√2²
This more complex example showcases operations involving roots of different orders, demonstrating how to simplify and combine them.
The page continues with several more intricate examples, each building on the complexity of the previous ones:
Example: 4√2 - 3√2² - √8² = 4√2 - ³√2² - 2√2 = 2√2 - ³√2²
This example combines subtraction of roots with different degrees and simplification.
Example: √50¹ - 3√2² - 5√5 - ³√51² = 5√2 - 5√5 - 3²³√2² - 3²³√2² = 5√2 - 5√5 - 6³√2
This complex example involves dodawanie i odejmowanie pierwiastków of various degrees, showcasing the steps needed to simplify such expressions.
Highlight: The examples progress from simple to complex, allowing students to gradually build their understanding of działania na pierwiastkach.
The final example on the page is particularly challenging:
Example: √√32² + ³√125¹ + 4√2 + 8√√2 = 2√√2 - 4²√2 - 9³√2 - 8³√2² = -7³√2² - 2²³√2
This example combines dodawanie pierwiastków o różnych podstawach and subsequent simplification, demonstrating the full range of skills needed for complex root arithmetic.
Vocabulary: Radicand - the number under the root symbol in a radical expression.
These examples collectively illustrate the importance of understanding the rules governing dodawanie i odejmowanie pierwiastków, as well as the ability to manipulate and simplify complex root expressions.
