prove that √3 and √5 is an irrational number

Question

prove that √3 and √5 is an irrational number

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

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    0
    2021-08-12T13:22:44+00:00

    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 is an contradiction.

    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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    0
    2021-08-12T13:22:48+00:00

    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

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