MATHCOUNTS State Sprint 2025 (Problem 26)Expressions & IdentitiesMediumWhen 7+43−7−43\sqrt{7+4\sqrt{3}}-\sqrt{7-4\sqrt{3}}7+43−7−43 is written in the form a3a\sqrt{3}a3, what is the value of aaa?Related TopicsCoreApproaches to Solving Square Root ProblemsAlgebraic IdentitiesCheck AnswerYour answer:CheckHints (4)Hint 17+43−7−43=a3\sqrt{7+4\sqrt{3}}-\sqrt{7-4\sqrt{3}}=a\sqrt{3}7+43−7−43=a3Hint 2Square both sides.Hint 37+43+7−43−2(7+43)(7−43)=3a27+4\sqrt{3}+7-4\sqrt{3}-2\sqrt{(7+4\sqrt{3})(7-4\sqrt{3})}=3a^27+43+7−43−2(7+43)(7−43)=3a27+43+7−43−249−48=3a27+4\sqrt{3}+7-4\sqrt{3}-2\sqrt{49-48}=3a^27+43+7−43−249−48=3a214−2=3a2⟹a=±214-2=3a^2\Longrightarrow a=\pm214−2=3a2⟹a=±2Hint 4a3=7+43−7−43>0a\sqrt{3}=\sqrt{7+4\sqrt{3}}-\sqrt{7-4\sqrt{3}}>0a3=7+43−7−43>0⟹a=2\Longrightarrow a=2⟹a=2Final Answer222Related Problems (4)AMC 10A 2020 (Problem 5)MathCounts 2026 Chapter Sprint Round (Problem 30)AMC 10A/12A 2023 (Problem 22/18)BMO1 2016/2017 (Problem 3)