Question

Show that √5+√7 is an irrational number.​

Answer 1

Step-by-step explanation:

Given :-

√5+√7

To find:-

√5+√7 is an Irrational number.

Solution :-

Given number = √5+√7

Let us assume that √5+√7 is a rational number.

√5+√7 is in the form of p/q,where p and q are integers and q ≠ 0

Let √5+√7 = a/b , a and b are co-primes

=> √7 = (a/b)-√5

On sharing both sides then

=> (√7)² = [(a/b)-√5]²

=> 7 = (a/b)²+(√5)²-2(a/b)(√5)

=> 7 = (a²/b²)+5-(2√5a/b)

Since, (x-y)² = x²-2xy+y²

=> 7-5 = (a²/b²)-(2√5a/b)

=> 2 = (a²-2√5ab)/b²

=> 2b² = a²-2√5ab

=> 2b²-a² = -2√5ab

=> a²-2b² = 2√5ab

=> (a²-2b²)/(2ab) = √5

=> √5 = (a²-2b²)/(2ab)

=> √5 is in the form of p/q

=> √5 is a rational number.

But √5 is not a rational number.

It is an irrational number.

This contradicts to our assumption that is √5+√7 is a rational number.

Therefore, √5+√7 is an irrational number.

Hence, Proved.

Used Method:-

→ Method of contradiction (Indirect method)

Note :-

→ If p is a rational number and q is an irrational number then

• p+q is an irrational number.
• p-q is an irrational number.
• pq is an irrational number.
• p/q is an irrational number.

→ The sum of two irrational numbers is also an irrational number.

√5 is an irrational number

√7 is an irrational number

Their sum √5+√7 is also an irrational number.

Answer 2

Step-by-step explanation:

Step-by-step explanation:

Given :-

√5+√7

To find:-

√5+√7 is an Irrational number.

Solution :-

Given number = √5+√7

Let us assume that √5+√7 is a rational number.

√5+√7 is in the form of p/q,where p and q are integers and q ≠ 0

Let √5+√7 = a/b , a and b are co-primes

=> √7 = (a/b)-√5

On sharing both sides then

=> (√7)² = [(a/b)-√5]²

=> 7 = (a/b)²+(√5)²-2(a/b)(√5)

=> 7 = (a²/b²)+5-(2√5a/b)

Since, (x-y)² = x²-2xy+y²

=> 7-5 = (a²/b²)-(2√5a/b)

=> 2 = (a²-2√5ab)/b²

=> 2b² = a²-2√5ab

=> 2b²-a² = -2√5ab

=> a²-2b² = 2√5ab

=> (a²-2b²)/(2ab) = √5

=> √5 = (a²-2b²)/(2ab)

=> √5 is in the form of p/q

=> √5 is a rational number.

But √5 is not a rational number.

It is an irrational number.

This contradicts to our assumption that is √5+√7 is a rational number.

Therefore, √5+√7 is an irrational number.

Hence, Proved.

Used Method:-

→ Method of contradiction (Indirect method)

Note :-

→ If p is a rational number and q is an irrational number then

p+q is an irrational number.

p-q is an irrational number.

pq is an irrational number.

p/q is an irrational number.

→ The sum of two irrational numbers is also an irrational number.

√5 is an irrational number

√7 is an irrational number

Their sum √5+√7 is also an irrational number.

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