What is (1+isqrt(3))^{10} ?

The solution to (1+isqrt(3))^{10} is -512-512sqrt(3)i

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simplify (1+isqrt(3))^{10}

simplify

(1 + i 3)^{10}

- 512 - 5123 i

Solution steps

(1 + i 3)^{10}

Apply binomial theorem:

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

a = 1, b = i 3

= i = 0 \sum 10 (i 10) \cdot 1^{(10 - i)} (i 3)^{i}

Expand summation

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

Simplify

\frac{10 !}{0 ! ( 10 - 0 ) !} \cdot 1^{10} (i 3)^{0} : 1

Simplify

\frac{10 !}{1 ! ( 10 - 1 ) !} \cdot 1^{9} (i 3)^{1} : 103 i

Simplify

\frac{10 !}{2 ! ( 10 - 2 ) !} \cdot 1^{8} (i 3)^{2} : 135 i^{2}

Simplify

\frac{10 !}{3 ! ( 10 - 3 ) !} \cdot 1^{7} (i 3)^{3} : 3603 i^{3}

Simplify

\frac{10 !}{4 ! ( 10 - 4 ) !} \cdot 1^{6} (i 3)^{4} : 1890 i^{4}

Simplify

\frac{10 !}{5 ! ( 10 - 5 ) !} \cdot 1^{5} (i 3)^{5} : 22683 i^{5}

Simplify

\frac{10 !}{6 ! ( 10 - 6 ) !} \cdot 1^{4} (i 3)^{6} : 5670 i^{6}

Simplify

\frac{10 !}{7 ! ( 10 - 7 ) !} \cdot 1^{3} (i 3)^{7} : 32403 i^{7}

Simplify

\frac{10 !}{8 ! ( 10 - 8 ) !} \cdot 1^{2} (i 3)^{8} : 3645 i^{8}

Simplify

\frac{10 !}{9 ! ( 10 - 9 ) !} \cdot 1^{1} (i 3)^{9} : 8103 i^{9}

Simplify

\frac{10 !}{10 ! ( 10 - 10 ) !} \cdot 1^{0} (i 3)^{10} : 243 i^{10}

= 1 + 103 i + 135 i^{2} + 3603 i^{3} + 1890 i^{4} + 22683 i^{5} + 5670 i^{6} + 32403 i^{7} + 3645 i^{8} + 8103 i^{9} + 243 i^{10}

Simplify

1 + 103 i + 135 i^{2} + 3603 i^{3} + 1890 i^{4} + 22683 i^{5} + 5670 i^{6} + 32403 i^{7} + 3645 i^{8} + 8103 i^{9} + 243 i^{10} : - 5123 i - 512

= - 5123 i - 512

Rewrite in standard complex form:

- 512 - 5123 i

= - 512 - 5123 i

(1 + 4 x)^{2}

(3 x^{2} - \frac{1}{10}) (3 x^{2} + \frac{1}{10})

2 (1 - cos (x))

e^{(x + 1)^{2}}

(\frac{3}{c} + \frac{c ^{2}}{4})^{7}

What is (1+isqrt(3))^{10} ?
The solution to (1+isqrt(3))^{10} is -512-512sqrt(3)i