Question

Prove that 2 V3 - 1 is an irrational number.
Prove that 7 - 2 v3 is an irrational number.
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Answer 1

Answer:

Step-by-step explanation:

prove thar 2√3 -1 is irrational

let √3 be a rational no.

√3 = a/b , where a,b not equal to 0, a&b are co- prime nos.

squaring both sides

3 = a²/b²

3b²= a² ---------- 1

b² = a²/3  ( 3 divides a², 3 divides a)

let a = 3c

squaring both sides

a² = 9c²

from 1

3b² = 9c²

b² = 3c²

b²/3 = c²     ( 3 divides b2, 3 divides b)

here, 3 is the common factor of  both a & b, which contradicts the statement that a&b are co-prime. so, √3 is irrational no.

let 2√3-1 be rational

2√3-1 = p/q , where p and q are integers.

2√3 = p/q+1

2√3 = p+q/q

√3 = p+q/2q

here, p,q,2 all are integers which means √3 is also an integer, which is not true. so, 2√3-1 is an irrational no.

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prove √3 irrational in the same way given above

let 7-2√3 be rational

7-2√3 = p/q , where p and q are integers

-2√3 = p/q-7

-2√3 = p-7q/q

√3 = p-7q/-2q

here, p,q,7,2 all are int. which means √3 is also an int. which is not true

so, 7-2√3 is an irrational no.

∴ HOPE IT WILL HELP YOU OUT∴