Question

Express $(1+i{)}^{10}$ in the standard form a + ib.

• A. 16i
• B. 32i
• C. 64i
• D. 32 + 32i

Answer 1

The correct option is B

32i

Since, given complex number has degree 10.
We need to use de moivre's theorem to remove 10th degree.
De Moivre's theorem used only in polar form so, express1+ i in polar form $1+i=\sqrt{2}(cos\frac{\pi }{4}+isin\frac{\pi }{4})$
$(1+i{)}^{10}$ = ${\left[\sqrt{2}\right(cos\frac{\pi }{4}+isin\frac{\pi }{4}]}^{10}$
Now use De Moivre's theorem
= $2^{\frac{10}{2}}[cos\left(\frac{10\pi }{4}\right)+isin\left(\frac{10\pi }{4}\right)]$
=$32[cos\frac{5\pi }{2}+isin\frac{5\pi }{2}]$
=$32[cos(2\pi +\frac{\pi }{2})+isin(2\pi +\frac{\pi }{2})]$
=$32[cos\frac{\pi }{2}+isin\frac{\pi }{2}]$
=32 (0+i) =32i