Questions and Answers
What is the correct simplified form of the surd √63?
जब दो सुर्ड्स के समान मूल हैं, तो उनके गुणांक क्या होते हैं?
सुर्ड्स के सimplification के लिए क्या प्रयोग किया जाता है?
निम्नलिखित में कौन सा सुर्ड्स के गुणन के नियम का उदाहरण है?
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सूचकांक के पहले नियम के अनुसार, क्या होता है?
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निम्नलिखित में कौन सा सुर्ड्स के सimplification के लिए प्रयोग किया जाता है?
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Study Notes
Simplifying Surds
Surds are irrational numbers that cannot be simplified into whole numbers or integers. They are the square roots of numbers that cannot be expressed as a ratio of two integers. Simplifying surds involves rewriting them in their simplest form by extracting square factors from underneath the root sign.
Factors of Surds
The process of simplifying surds starts by finding the factors of the number under the root sign. In the case of surds, these factors can be either rational or irrational. For example, consider the surd (\sqrt{63}). To simplify it, you would separate it into its prime factors, which in this case are (\sqrt{3}) and (\sqrt{7}).
Laws of Surds
There are three important laws of surds that can help in simplifying surds:

Rewrite the surd as a product of a square number and another number: If the number under the root sign has square factors, you can rewrite the surd as a product of this square number and another number. For example, (\sqrt{9} = 3).

Evaluate the root of the square number: Once you have rewritten the surd as a product of a square number and another number, evaluate the root of the square number. For example, (\sqrt{9} = 3).

Repeat if the number under the root still has square factors: If the number under the root still has square factors, repeat the process until there are no more square factors left.
Example: Simplifying (\sqrt{63})
Let's simplify (\sqrt{63}) using the laws of surds:
 Separate (\sqrt{63}) into its prime factors: (\sqrt{3}) and (\sqrt{7}).
 Rewrite (\sqrt{63}) as a product of these square numbers: (\sqrt{9}) and (\sqrt{7}).
 Evaluate the root of these square numbers: (3\sqrt{7}).
So, the simplified form of (\sqrt{63}) is (3\sqrt{7}).
Exercise
Simplify the following surds:
 (\sqrt{32})
 (\sqrt{40})
 (\sqrt{625})
For each surd, follow the steps outlined above to simplify it.
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Description
Test your understanding of simplifying surds, a fundamental concept in mathematics. This quiz will help you practice simplifying surds using the laws of surds and extracting square factors. Try to simplify various surds and evaluate your knowledge.