prove that √3 + √5 is an irrational number.
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Answers
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Let √3 + √5 be a rational number , say r
then √3 + √5 = r
On squaring both sides,
(√3 + √5)2 = r²
3 + 2 √15 + 5 = r²
8 + 2 √15 = r²
2 √15 = r² - 8
√15 = (r² - 8) / 2
Now (r² - 8) / 2 is a rational number and √15 is an irrational number .
Since a rational number cannot be equal to an irrational number . Our assumption that √3 + √5 is rational wrong
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Heya!
Here is ur answer...
Let us assume, √3+√5 is rational
And,
√3+√5 = a/b [where, a, b € Z]
√5 = a/b-√3
by squaring on both sides,
(√5)² = (a/b-√3)²
5 = a²/b² +3-2√2•a/b
2√3•a/b = a²/b² +3-5
2√3•a/b = a²/b²-2
2√3•a/b = a²-2b²/b²
2√3•a = a²-2b²/b
√3 = a²-2b²/2ab
Here, LHS is irrational
But, RHS is rational!
A rational and irrational are never equal!
Since, our assumption is false.
Therefore, √3+√5 is irrational
Hence preoved!