Question

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

Answer 1

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.