Question

prove that √3 + √5 is an irrational number.

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Answer 1

HeRe Is Your Ans ⤵

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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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Answer 2

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!