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

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Solution

$(\sqrt{2} + 2)^{2}$
$= (\sqrt{2})^{2} + 2 \times \sqrt{2} \times 2 + (2)^{2}$
$= 2 + 4 \sqrt{2} + 4 = 6 + 4 \sqrt{2}$ .
Here, $6$ is rational but $4 \sqrt{2}$ is irrational.
Since sum of rational and irrational number is an irrational number, $(\sqrt{2} + 2)^{2}$ is irrational number.

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