Question

Prove that the following are irrational:

(ii) $7 square root of 5$

Solution

(ii) $7 square root of 5$
Let us assume to the contrary, that $7 square root of 5$ is rational.
So we can find coprime integers a and b ( $space space not equal to$ 0 ) such that
$italic space italic space italic space italic space italic space italic space italic 7 square root of italic 5 italic equals a over b italic rightwards double arrow italic space italic space italic space italic space square root of italic 5 italic equals fraction numerator a over denominator italic 7 b end fraction$
Since, a and b are integers, $fraction numerator straight a over denominator 7 straight b end fraction$ is rational. and so, $square root of 5$ is rational
But this contradicts the fact that $square root of 5$ is irrational.
Therefore $7 square root of 5$ is irrational.

Some More Questions From Real Numbers Chapter

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Real Numbers

Prove that the following are irrational:

(ii) $7 square root of 5$

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

Entrance Exams Preparation

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Real Numbers

Prove that the following are irrational:(ii)

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

Prove that the following are irrational:

(ii) $7 square root of 5$