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Popular >

simplify (1+sqrt(3)i)^6

Solution

simplify

(1 + 3 i)^{6}

Solution

64

Solution steps

(1 + 3 i)^{6}

Apply binomial theorem:

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

a = 1, b = 3 i

= i = 0 \sum 6 (i 6) \cdot 1^{(6 - i)} (3 i)^{i}

Expand summation

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

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} \cdot 1^{6} (3 i)^{0} : 1

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} \cdot 1^{5} (3 i)^{1} : 63 i

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} \cdot 1^{4} (3 i)^{2} : 45 i^{2}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} \cdot 1^{3} (3 i)^{3} : 603 i^{3}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} \cdot 1^{2} (3 i)^{4} : 135 i^{4}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} \cdot 1^{1} (3 i)^{5} : 543 i^{5}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} \cdot 1^{0} (3 i)^{6} : 27 i^{6}

= 1 + 63 i + 45 i^{2} + 603 i^{3} + 135 i^{4} + 543 i^{5} + 27 i^{6}

Simplify

1 + 63 i + 45 i^{2} + 603 i^{3} + 135 i^{4} + 543 i^{5} + 27 i^{6} : 64

= 64

Popular Examples

2 x + 6 = x + 10

x^{2} + 6 x + 9 = 12

1^2+x^2=sqrt(2)^2

1^{2} + x^{2} = 2^{2}

x^{2} - 7 x + 18 = 0

x^{2} + 12 x + 36 = 0

Frequently Asked Questions (FAQ)

What is (1+sqrt(3)i)^6 ?
The solution to (1+sqrt(3)i)^6 is 64