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Popular >

simplify (1+i)^{12}

Solution

simplify

(1 + i)^{12}

Solution

- 64

Solution steps

(1 + i)^{12}

Apply binomial theorem:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = 1, b = i

= i = 0 \sum 12 (i 12) \cdot 1^{(12 - i)} i^{i}

Expand summation

= \frac{12 !}{0 ! ( 12 - 0 ) !} \cdot 1^{12} i^{0} + \frac{12 !}{1 ! ( 12 - 1 ) !} \cdot 1^{11} i^{1} + \frac{12 !}{2 ! ( 12 - 2 ) !} \cdot 1^{10} i^{2} + \frac{12 !}{3 ! ( 12 - 3 ) !} \cdot 1^{9} i^{3} + \frac{12 !}{4 ! ( 12 - 4 ) !} \cdot 1^{8} i^{4} + \frac{12 !}{5 ! ( 12 - 5 ) !} \cdot 1^{7} i^{5} + \frac{12 !}{6 ! ( 12 - 6 ) !} \cdot 1^{6} i^{6} + \frac{12 !}{7 ! ( 12 - 7 ) !} \cdot 1^{5} i^{7} + \frac{12 !}{8 ! ( 12 - 8 ) !} \cdot 1^{4} i^{8} + \frac{12 !}{9 ! ( 12 - 9 ) !} \cdot 1^{3} i^{9} + \frac{12 !}{10 ! ( 12 - 10 ) !} \cdot 1^{2} i^{10} + \frac{12 !}{11 ! ( 12 - 11 ) !} \cdot 1^{1} i^{11} + \frac{12 !}{12 ! ( 12 - 12 ) !} \cdot 1^{0} i^{12}

Simplify

\frac{12 !}{0 ! ( 12 - 0 ) !} \cdot 1^{12} i^{0} : 1

Simplify

\frac{12 !}{1 ! ( 12 - 1 ) !} \cdot 1^{11} i^{1} : 12 i

Simplify

\frac{12 !}{2 ! ( 12 - 2 ) !} \cdot 1^{10} i^{2} : 66 i^{2}

Simplify

\frac{12 !}{3 ! ( 12 - 3 ) !} \cdot 1^{9} i^{3} : 220 i^{3}

Simplify

\frac{12 !}{4 ! ( 12 - 4 ) !} \cdot 1^{8} i^{4} : 495 i^{4}

Simplify

\frac{12 !}{5 ! ( 12 - 5 ) !} \cdot 1^{7} i^{5} : 792 i^{5}

Simplify

\frac{12 !}{6 ! ( 12 - 6 ) !} \cdot 1^{6} i^{6} : 924 i^{6}

Simplify

\frac{12 !}{7 ! ( 12 - 7 ) !} \cdot 1^{5} i^{7} : 792 i^{7}

Simplify

\frac{12 !}{8 ! ( 12 - 8 ) !} \cdot 1^{4} i^{8} : 495 i^{8}

Simplify

\frac{12 !}{9 ! ( 12 - 9 ) !} \cdot 1^{3} i^{9} : 220 i^{9}

Simplify

\frac{12 !}{10 ! ( 12 - 10 ) !} \cdot 1^{2} i^{10} : 66 i^{10}

Simplify

\frac{12 !}{11 ! ( 12 - 11 ) !} \cdot 1^{1} i^{11} : 12 i^{11}

Simplify

\frac{12 !}{12 ! ( 12 - 12 ) !} \cdot 1^{0} i^{12} : i^{12}

= 1 + 12 i + 66 i^{2} + 220 i^{3} + 495 i^{4} + 792 i^{5} + 924 i^{6} + 792 i^{7} + 495 i^{8} + 220 i^{9} + 66 i^{10} + 12 i^{11} + i^{12}

Simplify

1 + 12 i + 66 i^{2} + 220 i^{3} + 495 i^{4} + 792 i^{5} + 924 i^{6} + 792 i^{7} + 495 i^{8} + 220 i^{9} + 66 i^{10} + 12 i^{11} + i^{12} : - 64

= - 64

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Frequently Asked Questions (FAQ)

What is (1+i)^{12} ?
The solution to (1+i)^{12} is -64