[tex]\sqrt{2-\sqrt{3+\sqrt{5-\sqrt{13+\sqrt{48}}}}} =?\\ \\\\ \sqrt{13+\sqrt{48}} = \sqrt{13+4\sqrt{3}} = \sqrt{(2\sqrt{3}+1)^2} = 2\sqrt{3}+1 \\ \sqrt{5-(2\sqrt{3}+1)} = \sqrt{4-2\sqrt{3}} = \sqrt{(\sqrt{3}-1)^2} = |\sqrt{3}-1| = \sqrt{3}-1 \\ \sqrt{3+(\sqrt{3}-1)} = \sqrt{2+\sqrt{3}}\\ \\ \sqrt{2+\sqrt{3}} = \sqrt{a}+\sqrt{b}\\ 2+\sqrt{3} = a+b+2\sqrt{ab}\\ \Rightarrow a+b = 2,\quad 4ab = 3 \\ \Rightarrow t^2-2t+\dfrac{3}{4} = 0 \Rightarrow 4t^2-8t+3 = 0\\\\\Delta = 64 - 4\cdot 3\cdot 3 = 64-36 = 28\neq p.p.[/tex]
Ultimul radical nu poate fi scris sub forma √a + √b, a,b ∈ ℚ.
Enunțul e greșit.
Cred că ai omis un factor la un radical.
