Math, asked by Rashivashist4020, 1 year ago

Given that sqrt 3 is irrational prove that 5+4 sqrt 3 is an irrational

Answers

Answered by Anonymous
0

Given :-

\sqrt{3} is an irrational number.

To prove :-

 5 + 4\sqrt{3} is an irrational number.

Proof :-

Let  5 + 4\sqrt{3} is a rational number.

Then, it can be expressed in the form of p/q where p and q are co primes and q\neq0 .

Since,

 5 + 4\sqrt{3}= \dfrac{p}{q}

transforming 5 on R. H. S

4\sqrt{3} = \dfrac{p}{q}-5

Also,

 \sqrt{3} = \dfrac{p}{4q}-5

we know that every operation of rational is an rational number.

\therefore Subtraction of rational is also rational.

But here, L. H. S is irrational since, 3 is an irrational number (given)

And R. H. S is rational number .

\therefore LHS ≠RHS

This contradiction is arisen due to our wrong assumption .

hence,  5 + 4\sqrt{3} is an irrational number proved...

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