Câu 18:

Tính:

a) \((\sqrt{3}+i)^{2}-(\sqrt{3}-i)^{2}\)

b) \((\sqrt{3}+i)^{2}+(\sqrt{3}-i)^{2}\)

c) \((\sqrt{3}+i)^{3}-(\sqrt{3}-i)^{3}\)

d) \(\frac{(\sqrt{3}+i)^{2}}{(\sqrt{3}-i)^{2}}\)

Lời giải:

a) \((\sqrt{3}+i)^{2}-(\sqrt{3}-i)^{2}\)

\(=[\sqrt{3}+i+\sqrt{3}-i][\sqrt{3}+i-\sqrt{3}+i]\)

\(=4 \sqrt{3} i\)

b) \((\sqrt{3}+i)^{2}+(\sqrt{3}-i)^{2}=3+2 \sqrt{3} i-1+3-2 \sqrt{3} i-1=4\)

c) \((\sqrt{3}+i)^{3}-(\sqrt{3}-i)^{3}\)

\(=[\sqrt{3}+i-\sqrt{3}+i]\cdot\left[(\sqrt{3}+i)^{2}+(\sqrt{3}-i)^{2}+(\sqrt{3}+i)(\sqrt{3}-i)\right]\)

\(=2 i(4+3+1)=2 i(4+4)=16 i\)

d) \(\frac{(\sqrt{3}+i)^{2}}{(\sqrt{3}-i)^{2}}=\frac{2+2 \sqrt{3} i}{2-2 \sqrt{3} i}\)

\(=\frac{1+\sqrt{3} i}{1-\sqrt{3} i}=\frac{(1+\sqrt{3} i)^{2}}{1+3}\)

\(=\frac{1+2 \sqrt{3} i-3}{4}=\frac{-1+\sqrt{3} i}{2}\)