Question

Cho \(\sqrt{\dfrac{7+3\sqrt{5}}{2}}=a+b\sqrt{5}\) (a,b ∈ R). Giá trị của biểu thức a + b là:

Answer 1

Ta có: 

\(\sqrt{\dfrac{7+3\sqrt{5}}{2}}\)

\(=\sqrt{\dfrac{2\cdot\left(7+3\sqrt{5}\right)}{2\cdot2}}\)

\(=\sqrt{\dfrac{14+6\sqrt{5}}{4}}\)

\(=\sqrt{\dfrac{\left(\sqrt{5}\right)^2+2\cdot\sqrt{5}\cdot3-3^2}{2^2}}\)

\(=\sqrt{\dfrac{\left(\sqrt{5}+3\right)^2}{2^2}}\)

\(=\dfrac{3+\sqrt{5}}{2}\)

Mà: \(\dfrac{3+\sqrt{5}}{2}=a+b\sqrt{5}\)

Nên:  \(\dfrac{3+\sqrt{5}}{2}=\dfrac{3}{2}+\dfrac{\sqrt{5}}{2}=\dfrac{3}{2}+\dfrac{1}{2}\sqrt{5}\)

Vậy: \(a=\dfrac{3}{2};b=\dfrac{1}{2}\)

\(\Rightarrow a+b=\dfrac{3}{2}+\dfrac{1}{2}=\dfrac{4}{2}=2\)

Answer 2

\(\sqrt{\dfrac{7+3\sqrt{5}}{2}}=\sqrt{\dfrac{14+6\sqrt{5}}{4}}=\sqrt{\left(\dfrac{3+\sqrt{5}}{2}\right)^2}\)

\(=\dfrac{3+\sqrt{5}}{2}\)

=>a=3/2; b=1/2

a+b=3/2+1/2=2