Question

prove that ✓11-✓6 is irrational​

Answer 1

let us root 11- root 6 is a rational number.

root 11 - root 6 =a/b

root 11 = a/b + root 6

squaring both side

11= (a/b)2 + 2a/b root6 + 6

2a/b root 6 = a2/b2 -5

root 6= a2-5b2/2ab

=> root 6 is a rational no.

This contradict the fact root 6 is irrational.

so,our assumption is wrong.

Hence,roo11 - root 6

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Answer 2

Answer:

√11-√6

let we assume that √11-√6 be the rational number so,

√11-√6=p/q

squaring on both sides

so,

(√11-√6)²=(p/q)²

by using (a-b)²=a²+b²-2ab formula

(√11)²+(√6)²-2.√11.√6=p²/q²

11+6-2√66=p²/q²

17-2√66=p²/q²

-2√66=p²/q²-17

-2√66=p²-17²q²/q²

√66=p²-17²q²/-2q²

L.H.S is irrational

R.H.S is rational

which is used by contradiction

so,

√11-√6 is irrational