Prove that 6 - √7 is an irrational number, given that √7 is an

Q) Prove that 6 – √7 is an irrational number, given that √7 is an irrational number.

Ans: Let us assume that 6 – √7 is a rational number

Let 6 – √7 = $\frac{p}{q}$ ; q ≠ 0 and p, q are integers

$\therefore$ √7 = $\frac{6q - p}{q}$

Since, p and q are integers; Therefore 6q – p is an integer

Therefore, $\frac{6q - p}{q}$ is a rational number

$\therefore$ √7 is a rational number

But it contradicts given condition that √7 is an irrational number

Therefore, 6 – √7 is an irrational number………… Hence Proved !

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