What is (1-isqrt(3))^5 ?

The solution to (1-isqrt(3))^5 is 16+16sqrt(3)i

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simplify (1-isqrt(3))^5

simplify

(1 - i 3)^{5}

16 + 163 i

Solution steps

(1 - i 3)^{5}

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 5 (i 5) \cdot 1^{(5 - i)} (- i 3)^{i}

Expand summation

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

Simplify

\frac{5 !}{0 ! ( 5 - 0 ) !} \cdot 1^{5} (- i 3)^{0} : 1

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} \cdot 1^{4} (- i 3)^{1} : - 53 i

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} \cdot 1^{3} (- i 3)^{2} : 30 i^{2}

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} \cdot 1^{2} (- i 3)^{3} : - 303 i^{3}

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} \cdot 1^{1} (- i 3)^{4} : 45 i^{4}

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} \cdot 1^{0} (- i 3)^{5} : - 93 i^{5}

= 1 - 53 i + 30 i^{2} - 303 i^{3} + 45 i^{4} - 93 i^{5}

Simplify

1 - 53 i + 30 i^{2} - 303 i^{3} + 45 i^{4} - 93 i^{5} : 163 i + 16

= 163 i + 16

Rewrite in standard complex form:

16 + 163 i

= 16 + 163 i

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\frac{( 1 + e ^{- 2 s} ) ^{2}}{s + 2}

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

\frac{\partial}{\partial y} (3 e^{y}) + e^{y} (5 x + 3 y)

e^{x} (1 - x)