Prove that
2 +  \sqrt{5 \:}
Is an irrational number

Question

Prove that
2 +  \sqrt{5 \:}
Is an irrational number

in progress 0
Madeline 3 months 2021-07-17T12:50:21+00:00 1 Answers 0 views 0

Answers ( )

    0
    2021-07-17T12:52:05+00:00

    Answer:

     \:

    Given: √2+√5

    We need to prove√2+√5 is an irrational number.

    Proof:

    Let us assume that √2+√5 is a rational number.

    A rational number can be written in the form of p/q where p,q are integers and q≠0

    √2+√5 = p/q

    On squaring both sides we get,

    (√2+√5)² = (p/q)²

    √2²+√5²+2(√5)(√2) = p²/q²

    2+5+2√10 = p²/q²

    7+2√10 = p²/q²

    2√10 = p²/q² – 7

    √10 = (p²-7q²)/2q

    p,q are integers then (p²-7q²)/2q is a rational number.

    Then √10 is also a rational number.

    But this contradicts the fact that √10 is an irrational number.

    Our assumption is incorrect

    √2+√5 is an irrational number.

    Hence proved..

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