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here is your question
⏩⏩ given that √3 is irrational sides... prove that (√5+√3) whole square is also irrational..... ⏪⏪
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Answers
Given :-
√3 is irrational
Here, we've to prove (√5 + √3)² is irrational.
Using identity (a + b)² = a² + 2ab + b²
a = √5 and b = √3
➡ (√5 + √3)² = (√5)² + 2(√5)(√3) + (√3)²
= 5 + 2(√5 × √3) + 3
= 5 + √15 + 3
= 8 + 2√15
We know that √15 is irrational.
And the product of an irrational number and rational number is always irrational.
➡ 2 × √15 = irrational
As a result, 2√15 is irrational and i.e 8 + 2√15 is also irrational since the sum of an irrational and a rational number is also irrational.
Hence Proved!
√3 is irrational
Here, we've to prove (√5 + √3)² is irrational.
Using identity (a + b)² = a² + 2ab + b²
a = √5 and b = √3
➡ (√5 + √3)² = (√5)² + 2(√5)(√3) + (√3)²
= 5 + 2(√5 × √3) + 3
= 5 + √15 + 3
= 8 + 2√15
We know that √15 is irrational.
And the product of an irrational number and rational number is always irrational.
➡ 2 × √15 = irrational
As a result, 2√15 is irrational and i.e 8 + 2√15 is also irrational since the sum of an irrational and a rational number is also irrational.
Hence Proved!