What is (sqrt(6)+i)^4 ?

The solution to (sqrt(6)+i)^4 is 1+20sqrt(6)i

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simplify (sqrt(6)+i)^4

simplify

(6 + i)^{4}

1 + 206 i

Solution steps

(6 + i)^{4}

Apply binomial theorem:

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

a = 6, b = i

= i = 0 \sum 4 (i 4) (6)^{(4 - i)} i^{i}

Expand summation

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

Simplify

\frac{4 !}{0 ! ( 4 - 0 ) !} (6)^{4} i^{0} : 36

Simplify

\frac{4 !}{1 ! ( 4 - 1 ) !} (6)^{3} i^{1} : 246 i

Simplify

\frac{4 !}{2 ! ( 4 - 2 ) !} (6)^{2} i^{2} : 36 i^{2}

Simplify

\frac{4 !}{3 ! ( 4 - 3 ) !} (6)^{1} i^{3} : 46 i^{3}

Simplify

\frac{4 !}{4 ! ( 4 - 4 ) !} (6)^{0} i^{4} : i^{4}

= 36 + 246 i + 36 i^{2} + 46 i^{3} + i^{4}

Simplify

36 + 246 i + 36 i^{2} + 46 i^{3} + i^{4} : 206 i + 1

= 206 i + 1

Rewrite in standard complex form:

1 + 206 i

= 1 + 206 i

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