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A student claims that \(5-\sqrt{7}\) is rational because 5 is rational and \(\sqrt{7}\) is a number. What is the error in the student's reasoning?एक विद्यार्थी दावा करता है कि \(5-\sqrt{7}\) परिमेय है, क्योंकि 5 परिमेय है और \(\sqrt{7}\) एक संख्या है। उसके तर्क में क्या त्रुटि है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: The sum or difference of a rational number and an irrational number is irrational.परिमेय संख्या में अपरिमेय संख्या जोड़ने या घटाने पर परिणाम अपरिमेय होता है।
If \(5-\sqrt{7}\) were rational, then \(5-(5-\sqrt{7})=\sqrt{7}\) would also be rational, which is impossible. Hence the difference is irrational. Exam tip: rational ± irrational is irrational.मान लें \(5-\sqrt{7}\) परिमेय हो। तब \(5-(5-\sqrt{7})=\sqrt{7}\) भी परिमेय हो जाएगा, जो असंभव है। अतः यह अंतर अपरिमेय है। परीक्षा में परिमेय ± अपरिमेय को अपरिमेय मानें।
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