Question

Express root x and root y as irrational

Answer 1

Answer: let root x be a rational number,p/q where p and q are integers

q is not 0 and p and q have no factor in common [except 1]

Step-by-step explanation:

root x=p/q

squaring both sides

x=p^2/q^2

xq^2=p^2

as q divides q^2,x dividesp^2

therefore x divides p

let p=x*k

[xk]^2=xq^2

x^2k^2/x=q^2

xk^2=q^2

as q divides xk^2,x divides q^2

therefore x divides q

thus p and q have a common factor 2 but 2 is prime

this contradicts the fact that p and q have no common factor[except 1]

therefore root x is not a rational number.

therefore root x is a rational number

Answer 2

let root x be a rational number,p/q where p and q are integers
q is not 0 and p and q have no factor in common [except 1]
Step-by-step explanation:
root x=p/q
squaring both sides
x=p^2/q^2
xq^2=p^2
as q divides q^2,x dividesp^2
therefore x divides p
let p=x*k
[xk]^2=xq^2
x^2k^2/x=q^2
xk^2=q^2
as q divides xk^2,x divides q^2
therefore x divides q
thus p and q have a common factor 2 but 2 is prime
this contradicts the fact that p and q have no common factor[except 1]
therefore root x is not a rational number.
therefore root x is a rational number.


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