Question

√5-√3 is a rational number?
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Answer 1

Answer:

no

Step-by-step explanation:

• 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

Answer 2

$\huge\sf\fbox\purple{ ♡ Solution ♡ }$

°°° Explanation °°°

Let 5 − 3

be a rational number of form ba

,where b =0

Squaring on both sides

( 5 − 3 ) 2

= ( ba ) 2

( 5 ) 2 +( 3 ) 2

−2( 5 )( 3 )= b 2

a 2 5+3+2 1 5= b 2

a 2 8+2 1 5= b 2

a 22 15 = b 2a 2 − 81

5 = 2b 2a 2 −8b 2 since 1

5 is irrational , 2b 2

a 2 −8b 2 is rational

Since LHS  = RHS, contradiction arises,

Therefore 5 − 3

is irrational