Question

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

Answer 1

Step-by-step explanation:

√3 + √5 = a/b

On squaring both sides we get,

(√3 + √5)² = (a/b)²

√3² + √5² + 2(√5)(√3) = a²/b²

3 + 5 + 2√15 = a²/b²

8 + 2√15 = a²/b²

2√15 = a²/b² – 8

√15 = (a²- 8b²)/2b

a, b are integers then (a²-8b²)/2b is a rational number.

Then √15 is also a rational number.

Answer 2

Given√3 + √5

To prove:√3 + √5 is an irrational number.

Let us assume that√3 + √5 is a rational number.

So it can be written in the form a/b

√3 + √5 = a/b

Here a and b are coprime numbers and b ≠ 0

Solving

√3 + √5 = a/b

On squaring both sides we get,

(√3 + √5)² = (a/b)²

√3² + √5² + 2(√5)(√3) = a²/b²

3 + 5 + 2√15 = a²/b²

8 + 2√15 = a²/b²

2√15 = a²/b² – 8

√15 = (a²- 8b²)/2b

a, b are integers then (a²-8b²)/2b is a rational number.

Then √15 is also a rational number.

But this contradicts the fact that √15 is an irrational number.

Our assumption is incorrect

√3 + √5 is an irrational number.

Hence, proved.

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