## prove that √3 and √5 is an irrational number

Question

prove that √3 and √5 is an irrational number

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1 month 2021-08-12T13:21:27+00:00 2 Answers 0 views 0

1. Step-by-step explanation:

let us suppose 3+root 5 is rational.

=>3+root 5 is in the form of p/q where p and q are integers and q is not =0

=>root5=p/q-3

​=>root 5=p-3q/q

as p, q and 3 are integers p-3q/3 is a rational number.

=>root 5 is a rational number.

but we know that root 5 is an irrational number.

this contradiction has arisen because of our wrong assumption that 3+root 5 is a rational number.

hence 3+ root 5 is an irrational number

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2. Step-by-step explanation:

Let √3+√5 be any rational number x

x=√3+√5

squaring both sides

x²=(√3+√5)²

x²=3+5+2√15

x²=8+2√15

x²-8=2√15

(x²-8)/2=√15

as x is a rational number so x²is also a rational number, 8 and 2 are rational nos. , so √15 must also be a rational number as quotient of two rational numbers is rational

but, √15 is an irrational number

so we arrive at a contradiction t

this shows that our supposition was wrong

so √3+√5 is not a rational number