Question

show that (√3 + 5)^2 is an irrational number

Answer 1

(√3+5)^2
(√3)^2+ 5^2 +2×√3×5
3+25+10√3
27+10√3
So, as we can see result is irrational.
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Answer 2

Solution :-

(√3+√5)² is a rational number,then there exists x and y co-prime integers such that,
(√3+√5)²=x/y

3+5+2√15=x/y

8+2√15=x/y

2√15=x/y-8

2√15=(x-8y)/y

√15=(x-8y)/2y

(x-8y)/2y is a rational
number.

Then √15 is also a rational number
But as we know √15 is an irrational number.

This contradiction has arisen is wrong.

(√3+√5)² is an irrational number.

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