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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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