Prove that √3 + √5 is irrational

Solution:

Considering √3 + √5 is rational.

Here √3 + √5 = a , where a is rational.

We can write it as

√3 = a - √5

By squaring both sides,

(√3)² = (a - √5)²

Using the algebraic identity (a - b)² = a² + b² - 2ab

3 = a² + 5 - 2a√5

2a√5 = a² + 2

Therefore, √5 = a² + 2/ 2a, which is a contradiction as the right hand side is a rational number while √5 is irrational

Therefore, √3 + √5 is irrational

✦ Try This: Prove that √3 is irrational

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 1

NCERT Exemplar Class 10 Maths Exercise 1.3 Problem 10

Prove that √3 + √5 is irrational

Summary:

Irrational numbers are those real numbers that cannot be represented in the form of a ratio. In other words, those real numbers that are not rational numbers are known as irrational numbers.√3 + √5 is irrational.

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