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

Given:

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

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,

$(2-\sqrt{2})(2+\sqrt{2})=(2)^2-(\sqrt{2})^2$                           [$(a+b)(a-b)=a^2-b^2$]

$=4-2$

$=2$

The decimal expansion of \( (2-\sqrt{2})(2+\sqrt{2}) \) is terminating.

Therefore, \( (2-\sqrt{2})(2+\sqrt{2}) \) is a rational number.   

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Updated on: 10-Oct-2022

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