The solution to (u+iv)^7 is (u^7-21u^5v^2+35u^3v^4-7uv^6)+(7u^6v-35u^4v^3+21u^2v^5-v^7)i

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simplify (u+iv)^7

Solution

simplify

(u + i v)^{7}

(u^{7} - 21 u^{5} v^{2} + 35 u^{3} v^{4} - 7 u v^{6}) + (7 u^{6} v - 35 u^{4} v^{3} + 21 u^{2} v^{5} - v^{7}) i

Solution steps

(u + i v)^{7}

Apply binomial theorem:

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

a = u, b = i v

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

Expand summation

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

Simplify

\frac{7 !}{0 ! ( 7 - 0 ) !} u^{7} (i v)^{0} : u^{7}

Simplify

\frac{7 !}{1 ! ( 7 - 1 ) !} u^{6} (i v)^{1} : 7 i u^{6} v

Simplify

\frac{7 !}{2 ! ( 7 - 2 ) !} u^{5} (i v)^{2} : 21 i^{2} u^{5} v^{2}

Simplify

\frac{7 !}{3 ! ( 7 - 3 ) !} u^{4} (i v)^{3} : 35 i^{3} u^{4} v^{3}

Simplify

\frac{7 !}{4 ! ( 7 - 4 ) !} u^{3} (i v)^{4} : 35 i^{4} u^{3} v^{4}

Simplify

\frac{7 !}{5 ! ( 7 - 5 ) !} u^{2} (i v)^{5} : 21 i^{5} u^{2} v^{5}

Simplify

\frac{7 !}{6 ! ( 7 - 6 ) !} u^{1} (i v)^{6} : 7 i^{6} u v^{6}

Simplify

\frac{7 !}{7 ! ( 7 - 7 ) !} u^{0} (i v)^{7} : i^{7} v^{7}

= u^{7} + 7 i u^{6} v + 21 i^{2} u^{5} v^{2} + 35 i^{3} u^{4} v^{3} + 35 i^{4} u^{3} v^{4} + 21 i^{5} u^{2} v^{5} + 7 i^{6} u v^{6} + i^{7} v^{7}

21 i^{2} u^{5} v^{2} = - 21 u^{5} v^{2}

35 i^{3} u^{4} v^{3} = - 35 i u^{4} v^{3}

35 i^{4} u^{3} v^{4} = 35 u^{3} v^{4}

i^{5} = i

7 i^{6} u v^{6} = - 7 u v^{6}

i^{7} = - i

= u^{7} + 7 i u^{6} v - 21 u^{5} v^{2} - 35 i u^{4} v^{3} + 35 u^{3} v^{4} + 21 i u^{2} v^{5} - 7 u v^{6} - i v^{7}

Rewrite in standard complex form:

(u^{7} - 21 u^{5} v^{2} + 35 u^{3} v^{4} - 7 u v^{6}) + (7 u^{6} v - 35 u^{4} v^{3} + 21 u^{2} v^{5} - v^{7}) i

= (u^{7} - 21 u^{5} v^{2} + 35 u^{3} v^{4} - 7 u v^{6}) + (7 u^{6} v - 35 u^{4} v^{3} + 21 u^{2} v^{5} - v^{7}) i

\frac{2 ( s ^{3} + 9 s ^{2} + 15 s + 5 )}{( s + 5 ) ^{2}}

\frac{x ^{3} - 9 x + c}{( x - 1 ) ^{2}}

\frac{( 1 + 2 cos ( x ) )}{( 2 + cos ( x ) ) ^{2}}

(2 nπ + π)^{2} (2 nπ - π)

What is (u+iv)^7 ?
The solution to (u+iv)^7 is (u^7-21u^5v^2+35u^3v^4-7uv^6)+(7u^6v-35u^4v^3+21u^2v^5-v^7)i