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Popular >

simplify (4+i)^4

Solution

simplify

(4 + i)^{4}

Solution

161 + 240 i

Solution steps

(4 + i)^{4}

Apply binomial theorem:

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

a = 4, b = i

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

Expand summation

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

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

Simplify

256 + 256 i + 96 i^{2} + 16 i^{3} + 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 (4+i)^4 ?
The solution to (4+i)^4 is 161+240i