Question

Prove that 7-4 root 7 is a irrational number

Answer 1

Step-by-step explanation:

Given:-

7-4 √7

To find:-

Prove that 7-4√7 is an irrational number?

Solution:-

Given number = 7-4√7

Let assume that 7-4√7 is a rational number.

It must be in the form of p/q, where p and q are integers and q≠0

Let 7-4√7 = a/b , where a and b are co-primes.

=> -4√7 = (a/b)-7

=> 4√7 = 7-(a/b)

=> 4√7 = (7b-a)/b

=> √7 = (7b-a)/4b

=> √7 is in the form of p/q

=> √7 is a rational number.

But √7 is an irrational number.

This contradicts to our assumption that 7-4√7 is a rational number.

It is an irrational number.

Hence , Proved.

Answer:-

7-4√7 is an irrational number.

Used Method:-

• Method of Contradiction (Indirect method)

Additional information:-

If q is a rational number and s is an irrational number then

• Sum of the rational and irrational = q+s is also an irrational number.

• Difference of the rational and irrational = q-s is also an irrational number.

• Product of the rational and irrational = qs is also an irrational number.

• Quotient of the rational and irrational = q/s is also an irrational number.