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

\frac{y ^{3} + 2 y ^{2} - 1}{y}

3 \cdot ((5, 10, 7) + (2, - 3, 7))

e^{2 jk w} (e^{4 jk w} + e^{- 4 jk w})

9 (sin (2 x) - cos (2 x))

\frac{( tan ( a ) + cot ( a ) ) sin ( a )}{csc ( a )}

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