Examine whether the following numbers are rational or irrational:$ (\sqrt{2}-2)^{2} $

Given:

\( (\sqrt{2}-2)^{2} \)

To do:

We have to classify the given number as rational or irrational.

Solution:  

A rational number can be expressed in either terminating decimal or non-terminating recurring decimals and an irrational number is expressed in non-terminating non-recurring decimals.

Therefore,

$(\sqrt{2}-2)^{2}=(\sqrt{2})^2+(2)^2-2\times\sqrt{2}\times2$

$=2+4-4\sqrt{2}$

$=6-4\sqrt{2}$

$\sqrt{2}=1.41421............$

The decimal expansion of \( \sqrt{2} \) is non-terminating and non-recurring.

The sum of a rational number and an irrational number is an irrational number.

Therefore, \( (\sqrt{2}-2)^{2} \) is an irrational number.  

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Updated on: 10-Oct-2022

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