Question

prove that 3 - 4 root 2 upon root 7 is irrational​

Answer 1

Step-by-step explanation:

let us assume that given no is rational

so, 3-4√2/7=a/b

3-4√2=7a/b

4√2=3-7a/b

√2=(3-7a/b)/4

but we know by our assumption that √2 is irrational.

so this no is irrational

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Answer 2

Step-by-step explanation:

Given :-

(3-4√2)/√7

To find:-

Prove that (3-4√2)/√7 is an irrational number?

Solution :-

Given number = (3-4√2)/√7

Let us assume that (3-4√2)/√7is a rational number

So this must be in the form of p/q

Where p and q are integers ,q≠0

Let (3-4√2)/√7 = a/b

Where a and b are co-primes

=> 3-4√2 = (a/b)×√7

=> 3-4√2 = (√7a/b)

On squaring both sides then

=> (3-4√2)² = (√7a/b)²

=> (3)²-2(3)(4√2)+(4√2)² = (√7a)²/(b)²

Since (a-b)² = a²-2ab+b²

=> 8-24√2+32 = 7a²/b²

=> 40-24√2 = 7a²/b²

=> 40-(7a²/b²) = 24√2

=> 24√2 = 40-(7a²/b²)

=> 24√2 = (40b²-7a²)/b²

=> √2 = (40b²-7a²)/(24b²)

=> √2 is in the form of p/q

=> √2 is a rational number (by the definition of rational number)

But √2 is not a rational number.

It is an irrational number.

This contradicts to our assumption that is

(3-4√2)/√7 is a rational number.

So, It is not a rational number

Therefore,(3-4√2)/√7 is an irrational number.

Hence, Proved.

Answer:-

(3-4√2)/√7 is an irrational number.

Used formulae:-

• The numbers in the form of p/q where p and q are integers and q≠0 are called Rational Numbers.

• The numbers not in the form of p/q where p and q are integers and q≠0 are called Irrational Numbers.

Used Method:-

• Method of Contradiction (Indirect method)