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Popular >

simplify (1+i)^6

Solution

simplify

(1 + i)^{6}

Solution

- 8 i

Solution steps

(1 + i)^{6}

Apply binomial theorem:

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

a = 1, b = i

= i = 0 \sum 6 (i 6) \cdot 1^{(6 - i)} i^{i}

Expand summation

= \frac{6 !}{0 ! ( 6 - 0 ) !} \cdot 1^{6} i^{0} + \frac{6 !}{1 ! ( 6 - 1 ) !} \cdot 1^{5} i^{1} + \frac{6 !}{2 ! ( 6 - 2 ) !} \cdot 1^{4} i^{2} + \frac{6 !}{3 ! ( 6 - 3 ) !} \cdot 1^{3} i^{3} + \frac{6 !}{4 ! ( 6 - 4 ) !} \cdot 1^{2} i^{4} + \frac{6 !}{5 ! ( 6 - 5 ) !} \cdot 1^{1} i^{5} + \frac{6 !}{6 ! ( 6 - 6 ) !} \cdot 1^{0} i^{6}

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} \cdot 1^{6} i^{0} : 1

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} \cdot 1^{5} i^{1} : 6 i

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} \cdot 1^{4} i^{2} : 15 i^{2}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} \cdot 1^{3} i^{3} : 20 i^{3}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} \cdot 1^{2} i^{4} : 15 i^{4}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} \cdot 1^{1} i^{5} : 6 i^{5}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} \cdot 1^{0} i^{6} : i^{6}

= 1 + 6 i + 15 i^{2} + 20 i^{3} + 15 i^{4} + 6 i^{5} + i^{6}

Simplify

1 + 6 i + 15 i^{2} + 20 i^{3} + 15 i^{4} + 6 i^{5} + i^{6} : - 8 i

= - 8 i

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Frequently Asked Questions (FAQ)

What is (1+i)^6 ?
The solution to (1+i)^6 is -8i