What is (sqrt(10)-i)^4 ?

The solution to (sqrt(10)-i)^4 is 41-36sqrt(10)i

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simplify (sqrt(10)-i)^4

simplify

(10 - i)^{4}

41 - 3610 i

Solution steps

(10 - i)^{4}

Apply binomial theorem:

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

a = 10, b = - i

= i = 0 \sum 4 (i 4) (10)^{(4 - i)} (- i)^{i}

Expand summation

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

Simplify

\frac{4 !}{0 ! ( 4 - 0 ) !} (10)^{4} (- i)^{0} : 100

Simplify

\frac{4 !}{1 ! ( 4 - 1 ) !} (10)^{3} (- i)^{1} : - 4010 i

Simplify

\frac{4 !}{2 ! ( 4 - 2 ) !} (10)^{2} (- i)^{2} : 60 i^{2}

Simplify

\frac{4 !}{3 ! ( 4 - 3 ) !} (10)^{1} (- i)^{3} : - 410 i^{3}

Simplify

\frac{4 !}{4 ! ( 4 - 4 ) !} (10)^{0} (- i)^{4} : i^{4}

= 100 - 4010 i + 60 i^{2} - 410 i^{3} + i^{4}

Simplify

100 - 4010 i + 60 i^{2} - 410 i^{3} + i^{4} : - 3610 i + 41

= - 3610 i + 41

Rewrite in standard complex form:

41 - 3610 i

= 41 - 3610 i

(5 X + 7 y)^{2}

a (g - h = m)

\frac{1}{k ( n - k )}

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

\frac{0.375 e ^{- 0.01 x}}{( 1 + 1.5 e ^{- 0.01 x} ) ^{2}}