Question

show that 5+√2 is a irrational number

Answer 1

Step-by-step explanation:

let us assume that 5+√3as rational number.

then, 5+√3=p/q

√3=p-5q/q

√3 is rational number.

It is contriduction to our assumption.

therefore 5+√3 is irrational

Answer 2

$\huge{\textbf{\underline{Irrational number!!}}}$

Let's know about irrational number first.

Irrational number :- A number which can not be expressed in the form of p/q where p and q are co - primes is called Irrational number.

Ex :- Every integer, fraction is a rational number.

We have to prove 5 + √2 is an irrational number.

Explanation :-

Let us assume 5 + √2 is a rational number.

• Then by definition of rational number.

We can expressed it is in the form of p/q.

$\dfrac{p}{q} = 5 + \sqrt{2}$

• Transforming 5 on Left hand side.

→ $\dfrac{p}{q}-5 = \sqrt{2}$

→ $\dfrac{p-5q}{q} = \sqrt{2}$ ---1)

As we know every operation of a rational number gives a rational number.

• That's why subtraction of rational is also rational.

Consider L.H.S part it is a rational number but R.H.S is not a rational number.

• From equation-1

Therefore, L.H.S ≠ R.H.S

our assumption is wrong because a rational number is never equal to an irrational number.

Hence, 5 + √2 is an irrational number.

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