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Popular >

simplify (1+2i)^{10}

Solution

simplify

(1 + 2 i)^{10}

Solution

237 - 3116 i

Solution steps

(1 + 2 i)^{10}

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 10 (i 10) \cdot 1^{(10 - i)} (2 i)^{i}

Expand summation

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

Simplify

\frac{10 !}{0 ! ( 10 - 0 ) !} \cdot 1^{10} (2 i)^{0} : 1

Simplify

\frac{10 !}{1 ! ( 10 - 1 ) !} \cdot 1^{9} (2 i)^{1} : 20 i

Simplify

\frac{10 !}{2 ! ( 10 - 2 ) !} \cdot 1^{8} (2 i)^{2} : 180 i^{2}

Simplify

\frac{10 !}{3 ! ( 10 - 3 ) !} \cdot 1^{7} (2 i)^{3} : 960 i^{3}

Simplify

\frac{10 !}{4 ! ( 10 - 4 ) !} \cdot 1^{6} (2 i)^{4} : 3360 i^{4}

Simplify

\frac{10 !}{5 ! ( 10 - 5 ) !} \cdot 1^{5} (2 i)^{5} : 8064 i^{5}

Simplify

\frac{10 !}{6 ! ( 10 - 6 ) !} \cdot 1^{4} (2 i)^{6} : 13440 i^{6}

Simplify

\frac{10 !}{7 ! ( 10 - 7 ) !} \cdot 1^{3} (2 i)^{7} : 15360 i^{7}

Simplify

\frac{10 !}{8 ! ( 10 - 8 ) !} \cdot 1^{2} (2 i)^{8} : 11520 i^{8}

Simplify

\frac{10 !}{9 ! ( 10 - 9 ) !} \cdot 1^{1} (2 i)^{9} : 5120 i^{9}

Simplify

\frac{10 !}{10 ! ( 10 - 10 ) !} \cdot 1^{0} (2 i)^{10} : 1024 i^{10}

= 1 + 20 i + 180 i^{2} + 960 i^{3} + 3360 i^{4} + 8064 i^{5} + 13440 i^{6} + 15360 i^{7} + 11520 i^{8} + 5120 i^{9} + 1024 i^{10}

Simplify

1 + 20 i + 180 i^{2} + 960 i^{3} + 3360 i^{4} + 8064 i^{5} + 13440 i^{6} + 15360 i^{7} + 11520 i^{8} + 5120 i^{9} + 1024 i^{10} : - 3116 i + 237

= - 3116 i + 237

Rewrite in standard complex form:

237 - 3116 i

= 237 - 3116 i

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Frequently Asked Questions (FAQ)

What is (1+2i)^{10} ?
The solution to (1+2i)^{10} is 237-3116i