Which option correctly describes the nature of (4+\sqrt{13})?
Answer and explanation
Correct answer: Irrational number
A rational number can be written as a fraction of integers, while an irrational number cannot be written in that form. The square root of a positive integer is irrational when the integer is not a perfect square. Since 13 is not a perfect square, \(\sqrt{13}\) is irrational. The number 4, however, is rational because it can be written as \(4/1\).
Adding a rational number to an irrational number always gives an irrational number. If the sum were rational, subtracting the rational number 4 would make \(\sqrt{13}\) rational, which is impossible. Therefore \(4+\sqrt{13}\) is irrational. It is consequently not an integer or a terminating decimal, so option A follows.
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What is the correct answer to this question?
Irrational number
Why is this the correct answer?
A rational number can be written as a fraction of integers, while an irrational number cannot be written in that form. The square root of a positive integer is irrational when the integer is not a perfect square. Since 13 is not a perfect square, \(\sqrt{13}\) is irrational. The number 4, however, is rational because it can be written as \(4/1\).
Adding a rational number to an irrational number always gives an irrational number. If the sum were rational, subtracting the rational number 4 would make \(\sqrt{13}\) rational, which is impossible. Therefore \(4+\sqrt{13}\) is irrational. It is consequently not an integer or a terminating decimal, so option A follows.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Polynomials. Topic: Irrational numbers and real numbers.
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