Question

Given that $\sqrt{2}$ is irrational, prove that $(5 + 3 \sqrt{2})$ is an irrational number.

Solution

Let us assume that $(5 + 3 \sqrt{2)}$ is rational. Then there exist co-prime positive integers a and b such that
$5 + 3 \sqrt{2} = \frac{a}{b} 3 \sqrt{2} = \frac{a}{b} - 5 \sqrt{2} = \frac{a - 5 b}{3 b} \Rightarrow \sqrt{2} i s r a t i o n a l [∵ a, b a r e i n t e g e r s, ∵ \frac{a - 5 b}{3 b} i s r a t i o n a l]$

This contradicts that $\sqrt{2}$ is irrational.
So our assumption is incorrect.

Hence, $(5 + 3 \sqrt{2})$ is an irrational number.

Some More Questions From Real Numbers Chapter

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Real Numbers

Given that $\sqrt{2}$ is irrational, prove that $(5 + 3 \sqrt{2})$ is an irrational number.

Some More Questions From Real Numbers Chapter

Express each number as a product of its prime factors: (iv) 5005

Express each number as a product of its prime factors: (v) 7429

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (i) 26 and 91

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (ii) 510 and 92

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (iii) 336 and 54

Find the LCM and HCF of the following integers by applying the prime factorisation method. (i) 12, 15 and 21

Find the LCM and HCF of the following integers by applying the prime factorisation method. (ii) 17, 23 and 29

Find the LCM and HCF of the following integers by applying the prime factorisation method. (iii) 8, 9 and 25

Check whether 6n can end with the digit 0 for any natural number n

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NCERT Sample Papers

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Real Numbers

Given that 2 is irrational, prove that (5 + 32) is an irrational number.

Some More Questions From Real Numbers Chapter

Express each number as a product of its prime factors: (iv) 5005

Express each number as a product of its prime factors: (v) 7429

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (i) 26 and 91

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (ii) 510 and 92

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (iii) 336 and 54

Find the LCM and HCF of the following integers by applying the prime factorisation method. (i) 12, 15 and 21

Find the LCM and HCF of the following integers by applying the prime factorisation method. (ii) 17, 23 and 29

Find the LCM and HCF of the following integers by applying the prime factorisation method. (iii) 8, 9 and 25

Check whether 6n can end with the digit 0 for any natural number n

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Given that $\sqrt{2}$ is irrational, prove that $(5 + 3 \sqrt{2})$ is an irrational number.