Exponential form: \(1 + i \sqrt{3} = \sqrt{3} e^{\pi i/3}\)
\((\sqrt{3} e^{\pi i/3})^6 = 27 e^{2 \pi i} = \boxed{27}\)
+2666 We have: \((1+ i \sqrt3)^6\)
Using the binomial theorem, we have: \(\large{{\sum\limits_{k=0}^n} {n \choose k} {a^{n-k}} b^k}\), where \(a=1\), \(b = i \sqrt3\), and \(n = 6\)
We have to evaluate all cases from when \(0 \le k \le 6\):
\(k = 0: {6 \choose 0} \times 1^6 \times{ i \sqrt3} ^0 = 1\)
\(k = 1: {6 \choose 1} \times 1^5 \times{ i \sqrt3} ^1 = 6i \sqrt 3 \)
\(k = 2: {6 \choose 2} \times 1^4 \times{ i \sqrt3} ^2 = -45\)
\(k = 3: {6 \choose 3} \times 1^3 \times{ i \sqrt3} ^3 = -60i \sqrt 3 \)
\(k = 4: {6 \choose 4} \times 1^2 \times{ i \sqrt3} ^4 = 135\)
\(k = 5: {6 \choose 5} \times 1^1 \times{ i \sqrt3} ^5 = 54i \sqrt 3 \)
\(k = 6: {6 \choose 6} \times 1^0 \times{ i \sqrt3} ^6 = -27\)
To find the answer, you must add all these numbers.
Hope this helps!!!
