What is (2+3i)^{10} ?

The solution to (2+3i)^{10} is -341525-145668i

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simplify (2+3i)^{10}

simplify

(2 + 3 i)^{10}

- 341525 - 145668 i

Solution steps

(2 + 3 i)^{10}

Apply binomial theorem:

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

a = 2, b = 3 i

= i = 0 \sum 10 (i 10) \cdot 2^{(10 - i)} (3 i)^{i}

Expand summation

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

Simplify

\frac{10 !}{0 ! ( 10 - 0 ) !} \cdot 2^{10} (3 i)^{0} : 1024

Simplify

\frac{10 !}{1 ! ( 10 - 1 ) !} \cdot 2^{9} (3 i)^{1} : 15360 i

Simplify

\frac{10 !}{2 ! ( 10 - 2 ) !} \cdot 2^{8} (3 i)^{2} : 103680 i^{2}

Simplify

\frac{10 !}{3 ! ( 10 - 3 ) !} \cdot 2^{7} (3 i)^{3} : 414720 i^{3}

Simplify

\frac{10 !}{4 ! ( 10 - 4 ) !} \cdot 2^{6} (3 i)^{4} : 1088640 i^{4}

Simplify

\frac{10 !}{5 ! ( 10 - 5 ) !} \cdot 2^{5} (3 i)^{5} : 1959552 i^{5}

Simplify

\frac{10 !}{6 ! ( 10 - 6 ) !} \cdot 2^{4} (3 i)^{6} : 2449440 i^{6}

Simplify

\frac{10 !}{7 ! ( 10 - 7 ) !} \cdot 2^{3} (3 i)^{7} : 2099520 i^{7}

Simplify

\frac{10 !}{8 ! ( 10 - 8 ) !} \cdot 2^{2} (3 i)^{8} : 1180980 i^{8}

Simplify

\frac{10 !}{9 ! ( 10 - 9 ) !} \cdot 2^{1} (3 i)^{9} : 393660 i^{9}

Simplify

\frac{10 !}{10 ! ( 10 - 10 ) !} \cdot 2^{0} (3 i)^{10} : 59049 i^{10}

= 1024 + 15360 i + 103680 i^{2} + 414720 i^{3} + 1088640 i^{4} + 1959552 i^{5} + 2449440 i^{6} + 2099520 i^{7} + 1180980 i^{8} + 393660 i^{9} + 59049 i^{10}

Simplify

1024 + 15360 i + 103680 i^{2} + 414720 i^{3} + 1088640 i^{4} + 1959552 i^{5} + 2449440 i^{6} + 2099520 i^{7} + 1180980 i^{8} + 393660 i^{9} + 59049 i^{10} : - 145668 i - 341525

= - 145668 i - 341525

Rewrite in standard complex form:

- 341525 - 145668 i

= - 341525 - 145668 i

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