Question

Daca a si b sunt numere rationale, astfel incat $a + b\sqrt{3} + 5=b+2a\sqrt{27}$, atunci produsul $a * b$ este egal cu:

Answer 1

$\it \sqrt{27}=\sqrt{9\cdot3}=3\sqrt3 \Rightarrow 2a\sqrt{27}=2a\cdot3\sqrt3=6a\sqrt3\\ \\ a+b\sqrt3+5=b+6a\sqrt3 \Rightarrow 5=b+6a\sqrt3-a-b\sqrt3 \Rightarrow \\ \\ \Rightarrow 5=(b-a)+\sqrt3(6a-b) \Rightarrow \begin{cases} \it b-a=5 \Rightarrow b=5+a\ \ \ (1)\\ \\ \it 6a-b=0 \Rightarrow b=6a\ \ \ \ \ \ (2)\end{cases}\\ \\ \\ (1),\ (2) \Rightarrow 6a=5+a \Rightarrow 6a-a=5 \Rightarrow 5a=5 \Rightarrow a=1\ \ \ \ \ (3)\\ \\ (2),\ (3) \Rightarrow b=6\cdot1 \Rightarrow b=6\\ \\ a\cdot b=1\cdot6=6$