Show that 5 + 3 2 is an irrational number given that 2 is an irrational number

Let us assume that 5 + 3 2 is a rational number, then 5 + 3 2 = r (say), where r is a rational number. ⇒ 3 2 = r - 5 ⇒ 2 = r - 5 3 As r is a rational number, so r - 5 3 is also rational number. ⇒ 2 is a rational number. But this contradicts that 2 is an irrational number. Hence, our assumption is wrong. Therefore, 5 + 3 2 is an irrational number.

Show that 3 + 2 5 is an irrational number.

Let us assume that 3 + 2 5 is a rational number, then 3 + 2 5 = r (say), where r is a rational number. ⇒ 2 5 = r - 3 ⇒ 5 = r - 3 2 As r is a rational number, so r - 3 2 is also rational number. ⇒ 5 is a rational number. But this contradicts that 5 is an irrational number. Hence, our assumption is wrong. Therefore, 3 + 2 5 is an irrational number.

Show that 3 + 2 3 is an irrational number.

Let us assume that 3 + 2 3 is a rational number, then 3 + 2 3 = r (say), where r is a rational number. ⇒ 2 3 = r - 3 ⇒ 3 = r - 3 2 As r is a rational number, so r - 3 2 is also rational number. ⇒ 3 is a rational number. But this contradicts that 3 is an irrational number. Hence, our assumption is wrong. Therefore, 3 + 2 3 is an irrational number.

Show that 2 + 3 5 is an irrational number.

Let us assume that 2 + 3 5 is a rational number, then 2 + 3 5 = r (say), where r is a rational number. ⇒ 2 + 3 = 5 r ⇒ 3 = 5 r - 2 As r is a rational number, so 5 r - 2 is also rational number. ⇒ 3 is a rational number. But this contradicts that 3 is an irrational number. Hence, our assumption is wrong. Therefore, 2 + 3 5 is an irrational number.

Show that 2 + 3 2 7 is an irrational number.

Let us assume that 2 + 3 2 7 is a rational number, then 2 + 3 2 7 = r (say), where r is a rational number. ⇒ 2 + 3 2 = 7 r ⇒ 3 2 = 7 r - 2 ⇒ 2 = 7 r - 2 3 As r is a rational number, so 7 r - 2 3 is also rational number. ⇒ 2 is a rational number. But this contradicts that 2 is an irrational number. Hence, our assumption is wrong. Therefore, 2 + 3 2 7 is an irrational number.

Show that 2 + 3 is an irrational number.

Let us assume that 2 + 3 is a rational number, then 2 + 3 = r (say), where r is a rational number. ⇒ 3 = r - 2 ⇒ 3 2 = r - 2 2 Use the identity a - b 2 = a 2 + b 2 - 2 a b ⇒ 3 = r 2 + 2 2 - 2 × r × 2 ⇒ 3 = r 2 + 2 - 2 r 2 ⇒ 2 r 2 = r 2 - 1 2 = r 2 - 1 2 r As r is a rational number and r ≠ 0 , so r 2 - 1 2 r is also rational number. ⇒ 2 is a rational number. But this contradicts that 2 is an irrational number. Hence, our assumption is wrong. Therefore, 2 + 3 is an irrational number.

Show that 1 3 - 5 is an irrational number.

Given, 1 3 - 5 Rationalise the denominator and apply the identity a - b a + b = a 2 - b 2 1 3 - 5 = 1 3 - 5 × 3 + 5 3 + 5 = 3 + 5 9 - 5 = 3 + 5 4 Let us assume that 1 3 - 5 is a rational number, then 3 + 5 = r (say), where r is a rational number. ⇒ 3 + 5 = r ⇒ 5 = r - 3 As r is a rational number and r ≠ 0 , so r - 3 is also rational number. ⇒ 5 is a rational number. But this contradicts that 5 is an irrational number. Hence, our assumption is wrong. Therefore, 1 3 - 5 is an irrational number.

Prove that 5 3 is an irrational number.

Suppose that 5 3 is a rational number, then 5 3 = p q , where p , q are both integers, q ≠ 0 and p , q have no common factors(except 1 ) ⇒ 5 = p q 3 ⇒ p 3 = 5 q 3 . . . . i As 5 divides 5 q 3 , so 5 divides p 3 but 5 is prime. ⇒ 5 divides p Let p = 5 m, where m is an integer. Substituting this value of p in i , we get 5 m 3 = 5 q 3 ⇒ 125 m 3 = 5 q 3 ⇒ 25 m 3 = q 3 As 5 divides 5 m 3 , so 5 divides q 3 but 5 is prime. ⇒ 5 divides q Thus, p and q have a common factor 5 .This contradicts that p a n d q have no common factors (except 1 ) Therefore, 5 is not a rational number. So, we conclude that 5 is an irrational number. Hence, 5 3 is an irrational number.

E M B I B E

CBSE Syllabus Standard Mathematics for Class X>Chapter 1 - Real Numbers>Exercise 1.3

M L Aggarwal Solutions for Chapter: Real Numbers, Exercise 3: Exercise 1.3

Author:M L Aggarwal

M L Aggarwal Mathematics Solutions for Exercise - M L Aggarwal Solutions for Chapter: Real Numbers, Exercise 3: Exercise 1.3

Attempt the practice questions on Chapter 1: Real Numbers, Exercise 3: Exercise 1.3 with hints and solutions to strengthen your understanding. CBSE Syllabus Standard Mathematics for Class X solutions are prepared by Experienced Embibe Experts.