    Question

# Comment on the type of number: 5√4

A
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B
Surd
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C
Irrational number
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D
Real number
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Solution

## The correct option is D Real numberCheck for Irrational number: Let's assume 5√4 is a rational number and its fractional form is pq, where p and q are integers, q≠0 and p and q have no common factors other than 1. 5√4=pq (raising fifth power on both sides) ⇒(415)5=p5q5⇒4q5=p5⇒2×2q5=p5 Here, p should be an even number as p5 is an even number. ∵even5=even number. ⇒p=2k (substituting this value in the original expression) 2×2q5=p5⇒2×2q5=(2k)5⇒2×2q5=32k5⇒q5=2×4k5 ∴q should also be an even number, which contradicts the assumption that p and q do not share any common factors other than 1. Hence, 5√4 is an irrational number. Check for Surd: nth root of a number which can not be simplified into a rational number is called surd. Here 5√4 can not be simplified into a rational number hence, it is a surd. Check for Quadratic Surd: Square root of any non-perfect square is called quadratic surd. 5√4 is not a quadratic surd as the 5th root is not a square root. Also, an irrational number is a real number.  Suggest Corrections  0      Similar questions
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