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Popular >

simplify (5+(2i)/n)^{10}

Solution

simplify

(5 + \frac{2 i}{n})^{10}

Solution

\frac{9765625 n ^{10} - 70312500 n ^{8} + 52500000 n ^{6} - 8400000 n ^{4} + 288000 n ^{2} - 1024}{n ^{10}} + \frac{100 ( 390625 n ^{8} - 750000 n ^{6} + 252000 n ^{4} - 19200 n ^{2} + 256 )}{n ^{9}} i

Solution steps

(5 + \frac{2 i}{n})^{10}

Apply binomial theorem:

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

a = 5, b = \frac{2 i}{n}

= i = 0 \sum 10 (i 10) \cdot 5^{(10 - i)} (\frac{2 i}{n})^{i}

Expand summation

= \frac{10 !}{0 ! ( 10 - 0 ) !} \cdot 5^{10} (\frac{2 i}{n})^{0} + \frac{10 !}{1 ! ( 10 - 1 ) !} \cdot 5^{9} (\frac{2 i}{n})^{1} + \frac{10 !}{2 ! ( 10 - 2 ) !} \cdot 5^{8} (\frac{2 i}{n})^{2} + \frac{10 !}{3 ! ( 10 - 3 ) !} \cdot 5^{7} (\frac{2 i}{n})^{3} + \frac{10 !}{4 ! ( 10 - 4 ) !} \cdot 5^{6} (\frac{2 i}{n})^{4} + \frac{10 !}{5 ! ( 10 - 5 ) !} \cdot 5^{5} (\frac{2 i}{n})^{5} + \frac{10 !}{6 ! ( 10 - 6 ) !} \cdot 5^{4} (\frac{2 i}{n})^{6} + \frac{10 !}{7 ! ( 10 - 7 ) !} \cdot 5^{3} (\frac{2 i}{n})^{7} + \frac{10 !}{8 ! ( 10 - 8 ) !} \cdot 5^{2} (\frac{2 i}{n})^{8} + \frac{10 !}{9 ! ( 10 - 9 ) !} \cdot 5^{1} (\frac{2 i}{n})^{9} + \frac{10 !}{10 ! ( 10 - 10 ) !} \cdot 5^{0} (\frac{2 i}{n})^{10}

Simplify

\frac{10 !}{0 ! ( 10 - 0 ) !} \cdot 5^{10} (\frac{2 i}{n})^{0} : 9765625

Simplify

\frac{10 !}{1 ! ( 10 - 1 ) !} \cdot 5^{9} (\frac{2 i}{n})^{1} : \frac{39062500 i}{n}

Simplify

\frac{10 !}{2 ! ( 10 - 2 ) !} \cdot 5^{8} (\frac{2 i}{n})^{2} : \frac{70312500 i ^{2}}{n ^{2}}

Simplify

\frac{10 !}{3 ! ( 10 - 3 ) !} \cdot 5^{7} (\frac{2 i}{n})^{3} : \frac{75000000 i ^{3}}{n ^{3}}

Simplify

\frac{10 !}{4 ! ( 10 - 4 ) !} \cdot 5^{6} (\frac{2 i}{n})^{4} : \frac{52500000 i ^{4}}{n ^{4}}

Simplify

\frac{10 !}{5 ! ( 10 - 5 ) !} \cdot 5^{5} (\frac{2 i}{n})^{5} : \frac{25200000 i ^{5}}{n ^{5}}

Simplify

\frac{10 !}{6 ! ( 10 - 6 ) !} \cdot 5^{4} (\frac{2 i}{n})^{6} : \frac{8400000 i ^{6}}{n ^{6}}

Simplify

\frac{10 !}{7 ! ( 10 - 7 ) !} \cdot 5^{3} (\frac{2 i}{n})^{7} : \frac{1920000 i ^{7}}{n ^{7}}

Simplify

\frac{10 !}{8 ! ( 10 - 8 ) !} \cdot 5^{2} (\frac{2 i}{n})^{8} : \frac{288000 i ^{8}}{n ^{8}}

Simplify

\frac{10 !}{9 ! ( 10 - 9 ) !} \cdot 5^{1} (\frac{2 i}{n})^{9} : \frac{25600 i ^{9}}{n ^{9}}

Simplify

\frac{10 !}{10 ! ( 10 - 10 ) !} \cdot 5^{0} (\frac{2 i}{n})^{10} : \frac{1024 i ^{10}}{n ^{10}}

= 9765625 + \frac{39062500 i}{n} + \frac{70312500 i ^{2}}{n ^{2}} + \frac{75000000 i ^{3}}{n ^{3}} + \frac{52500000 i ^{4}}{n ^{4}} + \frac{25200000 i ^{5}}{n ^{5}} + \frac{8400000 i ^{6}}{n ^{6}} + \frac{1920000 i ^{7}}{n ^{7}} + \frac{288000 i ^{8}}{n ^{8}} + \frac{25600 i ^{9}}{n ^{9}} + \frac{1024 i ^{10}}{n ^{10}}

Simplify

9765625 + \frac{39062500 i}{n} + \frac{70312500 i ^{2}}{n ^{2}} + \frac{75000000 i ^{3}}{n ^{3}} + \frac{52500000 i ^{4}}{n ^{4}} + \frac{25200000 i ^{5}}{n ^{5}} + \frac{8400000 i ^{6}}{n ^{6}} + \frac{1920000 i ^{7}}{n ^{7}} + \frac{288000 i ^{8}}{n ^{8}} + \frac{25600 i ^{9}}{n ^{9}} + \frac{1024 i ^{10}}{n ^{10}} : \frac{9765625 n ^{10} - 70312500 n ^{8} + 52500000 n ^{6} - 8400000 n ^{4} + 288000 n ^{2} - 1024}{n ^{10}} + \frac{100 ( 390625 n ^{8} - 750000 n ^{6} + 252000 n ^{4} - 19200 n ^{2} + 256 )}{n ^{9}} i

= \frac{9765625 n ^{10} - 70312500 n ^{8} + 52500000 n ^{6} - 8400000 n ^{4} + 288000 n ^{2} - 1024}{n ^{10}} + \frac{100 ( 390625 n ^{8} - 750000 n ^{6} + 252000 n ^{4} - 19200 n ^{2} + 256 )}{n ^{9}} i

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Frequently Asked Questions (FAQ)

What is (5+(2i)/n)^{10} ?
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