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Rational Numbers

Examine, whet...

Question

Examine, whether the following numbers are rational or irrational:
$(3 + \sqrt{3}) (3 - \sqrt{3})$ .

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Solution

Given, $(3 + \sqrt{3}) (3 - \sqrt{3})$ .
Since $(a + b) (a - b) = a^{2} - b^{2}$ ,
$(3 + \sqrt{3}) (3 - \sqrt{3})$ $= (3)^{2} - (\sqrt{3})^{3}$
$= 9 - 3 = 6$ .
Since $6$ is rational number, then $(3 + \sqrt{3}) (3 - \sqrt{3})$ is a rational number.

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