What is (sqrt(2)+sqrt(2)i)^7 ?

The solution to (sqrt(2)+sqrt(2)i)^7 is 64sqrt(2)-64sqrt(2)i

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simplify (sqrt(2)+sqrt(2)i)^7

simplify

(2 + 2 i)^{7}

642 - 642 i

Solution steps

(2 + 2 i)^{7}

Apply binomial theorem:

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

a = 2, b = 2 i

= i = 0 \sum 7 (i 7) (2)^{(7 - i)} (2 i)^{i}

Expand summation

= \frac{7 !}{0 ! ( 7 - 0 ) !} (2)^{7} (2 i)^{0} + \frac{7 !}{1 ! ( 7 - 1 ) !} (2)^{6} (2 i)^{1} + \frac{7 !}{2 ! ( 7 - 2 ) !} (2)^{5} (2 i)^{2} + \frac{7 !}{3 ! ( 7 - 3 ) !} (2)^{4} (2 i)^{3} + \frac{7 !}{4 ! ( 7 - 4 ) !} (2)^{3} (2 i)^{4} + \frac{7 !}{5 ! ( 7 - 5 ) !} (2)^{2} (2 i)^{5} + \frac{7 !}{6 ! ( 7 - 6 ) !} (2)^{1} (2 i)^{6} + \frac{7 !}{7 ! ( 7 - 7 ) !} (2)^{0} (2 i)^{7}

Simplify

\frac{7 !}{0 ! ( 7 - 0 ) !} (2)^{7} (2 i)^{0} : 82

Simplify

\frac{7 !}{1 ! ( 7 - 1 ) !} (2)^{6} (2 i)^{1} : 562 i

Simplify

\frac{7 !}{2 ! ( 7 - 2 ) !} (2)^{5} (2 i)^{2} : 1682 i^{2}

Simplify

\frac{7 !}{3 ! ( 7 - 3 ) !} (2)^{4} (2 i)^{3} : 2802 i^{3}

Simplify

\frac{7 !}{4 ! ( 7 - 4 ) !} (2)^{3} (2 i)^{4} : 2802 i^{4}

Simplify

\frac{7 !}{5 ! ( 7 - 5 ) !} (2)^{2} (2 i)^{5} : 1682 i^{5}

Simplify

\frac{7 !}{6 ! ( 7 - 6 ) !} (2)^{1} (2 i)^{6} : 562 i^{6}

Simplify

\frac{7 !}{7 ! ( 7 - 7 ) !} (2)^{0} (2 i)^{7} : 82 i^{7}

= 82 + 562 i + 1682 i^{2} + 2802 i^{3} + 2802 i^{4} + 1682 i^{5} + 562 i^{6} + 82 i^{7}

Simplify

82 + 562 i + 1682 i^{2} + 2802 i^{3} + 2802 i^{4} + 1682 i^{5} + 562 i^{6} + 82 i^{7} : 642 - 642 i

= 642 - 642 i

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What is (sqrt(2)+sqrt(2)i)^7 ?
The solution to (sqrt(2)+sqrt(2)i)^7 is 64sqrt(2)-64sqrt(2)i