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Popular >

simplify (i+1)^5

Solution

simplify

(i + 1)^{5}

Solution

- 4 - 4 i

Solution steps

(i + 1)^{5}

Apply binomial theorem:

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

a = i, b = 1

= i = 0 \sum 5 (i 5) i^{(5 - i)} \cdot 1^{i}

Expand summation

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

Simplify

\frac{5 !}{0 ! ( 5 - 0 ) !} i^{5} \cdot 1^{0} : i^{5}

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} i^{4} \cdot 1^{1} : 5 i^{4}

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} i^{3} \cdot 1^{2} : 10 i^{3}

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} i^{2} \cdot 1^{3} : 10 i^{2}

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} i^{1} \cdot 1^{4} : 5 i

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} i^{0} \cdot 1^{5} : 1

= i^{5} + 5 i^{4} + 10 i^{3} + 10 i^{2} + 5 i + 1

Simplify

i^{5} + 5 i^{4} + 10 i^{3} + 10 i^{2} + 5 i + 1 : - 4 i - 4

= - 4 i - 4

Rewrite in standard complex form:

- 4 - 4 i

= - 4 - 4 i

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Frequently Asked Questions (FAQ)

What is (i+1)^5 ?
The solution to (i+1)^5 is -4-4i