Solve the surd equation

Question

$$ \sqrt{14+ 8\sqrt{3}} = 2\sqrt{2} + b$$

Find b without using factoring

b=sqrt{6}

Answer 1

Other than saying that $b=\sqrt{14+8\sqrt{3}}-2\sqrt{2}$ is a perfectly correct answer, I will tell you that

$$ \sqrt{a+\sqrt{b}}=\sqrt{\frac{a+c}{2}}+\sqrt{\frac{a-c}{2}} $$ where $c^2=a^2-b$.