What is (sqrt(10)-i)^6 ?

The solution to (sqrt(10)-i)^6 is -351-406sqrt(10)i

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simplify (sqrt(10)-i)^6

simplify

(10 - i)^{6}

- 351 - 40610 i

Solution steps

(10 - i)^{6}

Apply binomial theorem:

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

a = 10, b = - i

= i = 0 \sum 6 (i 6) (10)^{(6 - i)} (- i)^{i}

Expand summation

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

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} (10)^{6} (- i)^{0} : 1000

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} (10)^{5} (- i)^{1} : - 60010 i

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} (10)^{4} (- i)^{2} : 1500 i^{2}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} (10)^{3} (- i)^{3} : - 20010 i^{3}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} (10)^{2} (- i)^{4} : 150 i^{4}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} (10)^{1} (- i)^{5} : - 610 i^{5}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} (10)^{0} (- i)^{6} : i^{6}

= 1000 - 60010 i + 1500 i^{2} - 20010 i^{3} + 150 i^{4} - 610 i^{5} + i^{6}

Simplify

1000 - 60010 i + 1500 i^{2} - 20010 i^{3} + 150 i^{4} - 610 i^{5} + i^{6} : - 40610 i - 351

= - 40610 i - 351

Rewrite in standard complex form:

- 351 - 40610 i

= - 351 - 40610 i

(4 + \frac{3 i}{n})^{2}

(7 x - \frac{2}{3}) (7 x + \frac{2}{3})

\frac{1}{3 ( 11 \cdot 4 ^{x} - 5 )}

\frac{200 e}{( 1 + e ) ^{2}}