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Popular >

simplify (1+4i)^4

Solution

simplify

(1 + 4 i)^{4}

Solution

161 - 240 i

Solution steps

(1 + 4 i)^{4}

Apply binomial theorem:

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

a = 1, b = 4 i

= i = 0 \sum 4 (i 4) \cdot 1^{(4 - i)} (4 i)^{i}

Expand summation

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

Simplify

\frac{4 !}{0 ! ( 4 - 0 ) !} \cdot 1^{4} (4 i)^{0} : 1

Simplify

\frac{4 !}{1 ! ( 4 - 1 ) !} \cdot 1^{3} (4 i)^{1} : 16 i

Simplify

\frac{4 !}{2 ! ( 4 - 2 ) !} \cdot 1^{2} (4 i)^{2} : 96 i^{2}

Simplify

\frac{4 !}{3 ! ( 4 - 3 ) !} \cdot 1^{1} (4 i)^{3} : 256 i^{3}

Simplify

\frac{4 !}{4 ! ( 4 - 4 ) !} \cdot 1^{0} (4 i)^{4} : 256 i^{4}

= 1 + 16 i + 96 i^{2} + 256 i^{3} + 256 i^{4}

Simplify

1 + 16 i + 96 i^{2} + 256 i^{3} + 256 i^{4} : - 240 i + 161

= - 240 i + 161

Rewrite in standard complex form:

161 - 240 i

= 161 - 240 i

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Frequently Asked Questions (FAQ)

What is (1+4i)^4 ?
The solution to (1+4i)^4 is 161-240i