Which statement is correct about ∛9?
Answer and explanation
Correct answer: It is irrational
The cube root of an integer is rational only when the number under the cube-root sign is a perfect cube of a rational number. The integer 9 is not a perfect cube: the nearby perfect cubes are 1³ = 1, 2³ = 8, and 3³ = 27. Therefore ∛9 cannot simplify to an integer. More formally, if ∛9 were rational, its reduced numerator and denominator would imply that 9 is a cube of an integer, which is impossible because its prime factorisation is 3² and the exponent 2 is not a multiple of 3. Hence ∛9 is irrational, so option A is correct. It is not 3 because 3³ = 27, and an irrational number cannot be rational or an integer.
Frequently asked questions
What is the correct answer to this question?
It is irrational
Why is this the correct answer?
The cube root of an integer is rational only when the number under the cube-root sign is a perfect cube of a rational number. The integer 9 is not a perfect cube: the nearby perfect cubes are 1³ = 1, 2³ = 8, and 3³ = 27. Therefore ∛9 cannot simplify to an integer. More formally, if ∛9 were rational, its reduced numerator and denominator would imply that 9 is a cube of an integer, which is impossible because its prime factorisation is 3² and the exponent 2 is not a multiple of 3. Hence ∛9 is irrational, so option A is correct. It is not 3 because 3³ = 27, and an irrational number cannot be rational or an integer.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Real Numbers. Topic: Irrational numbers.
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