Prove that the following numbers are irrational: (i) ∛2 (ii) ∛3 (iii) ∜5.

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KnowledgeBoat · Accepted Answer

(i) Suppose that = , where p, q are integers , q ≠ 0 , p and q have no common factors (except 1) As 2 divides 2q³ 2 divides p³ 2 divides p (using generalisation of theorem 1) Let p = 2k , where k is an integer. Substituting this value of p in (i), we get (2k)³ = 2q³ 8k³ = 2q³ 4k³ = q³ As 2 divides 4k³ 2 divides q³ 2 divides q (using generalisation of theorem 1) Thus, p and q have a common factor 2. This contradicts that p and q have no common factors (except 1). Hence, our supposition is wrong. It follows that cannot be expressed as , where p, q are integers, q > 0, p and q have no common factors (except 1). ∴ is an irrational number. (ii) Suppose that = , where p, q are integers, q ≠ 0, p and q have no common factors (except 1) As 3 divides 3q³ 3 divides p³ 3 divides p (using generalisation of theorem 1) Let p = 3k, where k is an integer. Substituting this value of p in (i), we get (3k)³ = 3q³ 27k³ = 3q³ 9k³ = q³ As 3 divides 9k³ 3 divides q³ 3 divides q (using generalisation of theorem 1) Thus, p and q have a common factor 3. This contradicts that p and q have no common factors (except 1). Hence, our supposition is wrong . It follows that cannot be expressed as , where p, q are integers, q > 0, p and q have no common factors (except 1). Therefore, is an irrational number. (iii) Suppose that = , where p, q are integers, q ≠ 0, p and q have no common factors (except 1) As 5 divides 5q⁴ 5 divides p⁴ 5 divides p (using generalisation of theorem 1) Let p= 5k, where k is an integer. Substituting this value of p in (i), we get (5k)⁴ = 5q⁴ 625k⁴ = 5q⁴ 125k⁴ = q⁴ As 5 divides 125k⁴ 5 divides q⁴ 5 divides q (using generalisation of theorem 1) Thus, p and q have a common factor 5. This contradicts that p and q have no common factors (except 1). Hence, our supposition is wrong . It follows that cannot be expressed as , where p, q are integers, q > 0, p and q have no common factors (except 1). Therefore, is an irrational number .

Mathematics

Prove that the following numbers are irrational: (i) ∛2 (ii) ∛3 (iii) ∜5.

Rational Irrational Nos

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