Question
Examine, whether the following numbers are rational or irrational:
$\big(2-\sqrt{2}\big)\big(2+\sqrt{2}\big)$
$\big(2-\sqrt{2}\big)\big(2+\sqrt{2}\big)$
✓
Answer
$\big(2-\sqrt{2}\big)\big(2+\sqrt{2}\big)$
We have,
$\big(2-\sqrt{2}\big)\big(2+\sqrt{2}\big)=(2)^2-(\sqrt{2})^2$[Since, $(a+b)(a-b)=a^2-b^2$]
$4-2=\frac{2}{1}$
Since, $2$ is a rational number.
$\big(2-\sqrt{2}\big)\big(2+\sqrt{2}\big)$ is a rational number.
We have,
$\big(2-\sqrt{2}\big)\big(2+\sqrt{2}\big)=(2)^2-(\sqrt{2})^2$[Since, $(a+b)(a-b)=a^2-b^2$]
$4-2=\frac{2}{1}$
Since, $2$ is a rational number.
$\big(2-\sqrt{2}\big)\big(2+\sqrt{2}\big)$ is a rational number.
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