6 under root 5 is irrational number prove it
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Let assume 6√5 is a rational number
6√5= p/q
√5=p/6q
Now again,
√5= a/b
on squaring to both the sides
(√5)^2= (a/b)^2
5= a^2/b^2
5b^2= a^2
b^2= a^2/5
By theorem 1.3, it divides a^2 so it will divides a also
on putting a= 5c
b^2= (5c)^2/5
b^2= 25c^2/5
b^2= 5c^2
b^2/5= c^2
Again, by theorem 1.3, it divides b^2 so it will divides b also
As we can see that the common factor is 5
Hence, our assumption is wrong as 6√5 is rational number
Therefore, 6√5 is an irrational number
HOPE it helps!!!!
6√5= p/q
√5=p/6q
Now again,
√5= a/b
on squaring to both the sides
(√5)^2= (a/b)^2
5= a^2/b^2
5b^2= a^2
b^2= a^2/5
By theorem 1.3, it divides a^2 so it will divides a also
on putting a= 5c
b^2= (5c)^2/5
b^2= 25c^2/5
b^2= 5c^2
b^2/5= c^2
Again, by theorem 1.3, it divides b^2 so it will divides b also
As we can see that the common factor is 5
Hence, our assumption is wrong as 6√5 is rational number
Therefore, 6√5 is an irrational number
HOPE it helps!!!!
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