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Question

Prove that the following are irrational:
(i) 12 (ii) 75 (iii) 6+2

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Solution

(i) 12

Let us assume 12 is rational.

So we can write this number as

12=ab ---- (1)

Here, a and b are two co-prime numbers and b is not equal to zero.

Simplify the equation (1) multiply by 2 both sides, we get

1=a2b

Now, divide by b, we get

b=a2 or ba=2

Here, a and b are integers so, ba is a rational number,

so 2 should be a rational number.

But 2 is a irrational number, so it is contradictory.

Therefore, 12 is irrational number.

(ii) 75

Let us assume 75 is rational.

So, we can write this number as

75=ab ---- (1)

Here, a and b are two co-prime numbers and b is not equal to zero.

Simplify the equation (1) divide by 7 both sides, we get

5=a7b

Here, a and b are integers, so a7b is a rational

number, so 5 should be a rational number.

But 5 is a irrational number, so it is contradictory.

Therefore, 75 is irrational number.

(iii) 6+2

Let us assume 6+2 is rational.

So we can write this number as

6+2=ab ---- (1)

Here, a and b are two co-prime number and b is not equal to zero.

Simplify the equation (1) subtract 6 on both sides, we get

2=ab6

2=a6bb

Here, a and b are integers so, a6bb is a rational
number, so 2 should be a rational number.

But 2 is a irrational number, so it is contradictory.

Therefore, 6+2 is irrational number.

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