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Popular >

simplify (i^{11}+i^{-5})^6

Solution

simplify

(i^{11} + i^{- 5})^{6}

Solution

- 64

Solution steps

(i^{11} + i^{- 5})^{6}

Apply binomial theorem:

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

a = i^{11}, b = i^{- 5}

= i = 0 \sum 6 (i 6) (i^{11})^{(6 - i)} (i^{- 5})^{i}

Expand summation

= \frac{6 !}{0 ! ( 6 - 0 ) !} (i^{11})^{6} (i^{- 5})^{0} + \frac{6 !}{1 ! ( 6 - 1 ) !} (i^{11})^{5} (i^{- 5})^{1} + \frac{6 !}{2 ! ( 6 - 2 ) !} (i^{11})^{4} (i^{- 5})^{2} + \frac{6 !}{3 ! ( 6 - 3 ) !} (i^{11})^{3} (i^{- 5})^{3} + \frac{6 !}{4 ! ( 6 - 4 ) !} (i^{11})^{2} (i^{- 5})^{4} + \frac{6 !}{5 ! ( 6 - 5 ) !} (i^{11})^{1} (i^{- 5})^{5} + \frac{6 !}{6 ! ( 6 - 6 ) !} (i^{11})^{0} (i^{- 5})^{6}

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} (i^{11})^{6} (i^{- 5})^{0} : i^{66}

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} (i^{11})^{5} (i^{- 5})^{1} : 6 i^{50}

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} (i^{11})^{4} (i^{- 5})^{2} : 15 i^{34}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} (i^{11})^{3} (i^{- 5})^{3} : 20 i^{18}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} (i^{11})^{2} (i^{- 5})^{4} : 15 i^{2}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} (i^{11})^{1} (i^{- 5})^{5} : \frac{6}{i ^{14}}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} (i^{11})^{0} (i^{- 5})^{6} : \frac{1}{i ^{30}}

= i^{66} + 6 i^{50} + 15 i^{34} + 20 i^{18} + 15 i^{2} + \frac{6}{i ^{14}} + \frac{1}{i ^{30}}

Simplify

i^{66} + 6 i^{50} + 15 i^{34} + 20 i^{18} + 15 i^{2} + \frac{6}{i ^{14}} + \frac{1}{i ^{30}} : - 64

= - 64

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Frequently Asked Questions (FAQ)

What is (i^{11}+i^{-5})^6 ?
The solution to (i^{11}+i^{-5})^6 is -64