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Popular >

simplify (-sqrt(3)+i)^6

Solution

simplify

(- 3 + i)^{6}

Solution

- 64

Solution steps

(- 3 + i)^{6}

Apply binomial theorem:

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

a = - 3, b = i

= i = 0 \sum 6 (i 6) (- 3)^{(6 - i)} i^{i}

Expand summation

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

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} (- 3)^{6} i^{0} : 27

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} (- 3)^{5} i^{1} : - 543 i

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} (- 3)^{4} i^{2} : 135 i^{2}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} (- 3)^{3} i^{3} : - 603 i^{3}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} (- 3)^{2} i^{4} : 45 i^{4}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} (- 3)^{1} i^{5} : - 63 i^{5}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} (- 3)^{0} i^{6} : i^{6}

= 27 - 543 i + 135 i^{2} - 603 i^{3} + 45 i^{4} - 63 i^{5} + i^{6}

Simplify

27 - 543 i + 135 i^{2} - 603 i^{3} + 45 i^{4} - 63 i^{5} + i^{6} : - 64

= - 64

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Frequently Asked Questions (FAQ)

What is (-sqrt(3)+i)^6 ?
The solution to (-sqrt(3)+i)^6 is -64