Question

Prove that √5-√3 is not a rational number.

Answer 1

Answer:

Let us assume that √5 - √3 is a rational number.

=> √5 - √3 =

Here .. a and b are co-prime numbers.

Now, squaring on both sides.

=> (√5 - √3)² =

(a + b)² = a² + b² + 2ab

=> (√5)² + (√3)² - 2(√5)(√3) =

=> 5 + 3 - 2√15 =

=> 8 - 2√15 =

=> - 2√15 =

=> √15 =

Here ...

is a rational number.

So, √15 is also a rational number. But we know that √15 is irrational number.

So, our assumption is wrong.

√5 - √3 is a irrational number

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