1.

Given that √3 is an irrational number, show that (5 +2√3) is an irrational number.

Answer»

Let 5 + 2√3 be a rational number.

5 + 2√3 = \(\frac{p}{q}\), where p and q are co-prime integers.

2√3 = \(\frac{p}{q}\) - 5

= \(\frac{p - 5q}{q}\)

√3 = \(\frac{p - 5q}{2q}\)

Here, \(\frac{p - 5q}{2q}\) is rational as p and q are integers. 

But it is given that √3 is irrational.

LHS is irrational and RHS is rational.

Which contradicts our assumption that 5 + 2√3 is a rational number.

5 + 2√3 is an irrational number.


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