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Popular >

simplify (1+2i)^6

Solution

simplify

(1 + 2 i)^{6}

Solution

117 + 44 i

Solution steps

(1 + 2 i)^{6}

Apply binomial theorem:

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

a = 1, b = 2 i

= i = 0 \sum 6 (i 6) \cdot 1^{(6 - i)} (2 i)^{i}

Expand summation

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

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} \cdot 1^{6} (2 i)^{0} : 1

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} \cdot 1^{5} (2 i)^{1} : 12 i

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} \cdot 1^{4} (2 i)^{2} : 60 i^{2}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} \cdot 1^{3} (2 i)^{3} : 160 i^{3}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} \cdot 1^{2} (2 i)^{4} : 240 i^{4}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} \cdot 1^{1} (2 i)^{5} : 192 i^{5}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} \cdot 1^{0} (2 i)^{6} : 64 i^{6}

= 1 + 12 i + 60 i^{2} + 160 i^{3} + 240 i^{4} + 192 i^{5} + 64 i^{6}

Simplify

1 + 12 i + 60 i^{2} + 160 i^{3} + 240 i^{4} + 192 i^{5} + 64 i^{6} : 44 i + 117

= 44 i + 117

Rewrite in standard complex form:

117 + 44 i

= 117 + 44 i

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Frequently Asked Questions (FAQ)

What is (1+2i)^6 ?
The solution to (1+2i)^6 is 117+44i