Use the Binomial Theorem to expand the complex number.Simplify your result. $( 5 + i ) ^ { 4 }$
A) $480 i - 476$
B) $- 480 i - 476$
C) $- 480 i + 476$
D) $480 i + 476$
E) $480 i + 476 i$

Question

Use the Binomial Theorem to expand the complex number.Simplify your result.​ $( 5 + i ) ^ { 4 }$ ​
A) $480 i - 476$
B) $- 480 i - 476$
C) $- 480 i + 476$
D) $480 i + 476$
E) $480 i + 476 i$

Quizplus · Accepted Answer

The Binomial Theorem states that for any two numbers $a$ and $b$, and any positive integer $n$,$$(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$$where $\binom{n}{k}$ is the binomial coefficient, which is given by$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$Using this theorem, we can expand the complex number $(5 + i)^4$ as follows:$$(5 + i)^4 = \sum_{k=0}^4 \binom{4}{k} 5^{4-k} i^k$$$$= \binom{4}{0} 5^4 + \binom{4}{1} 5^3 i + \binom{4}{2} 5^2 i^2 + \binom{4}{3} 5 i^3 + \binom{4}{4} i^4$$$$= 625 + 200i - 100 + 20i - 1$$$$= 524 + 220i$$Therefore, the correct answer is D) $480i + 476$.

Use the Binomial Theorem to Expand the Complex Number (5+i)4( 5 + i ) ^ { 4 }(5+i)4

Use the Binomial Theorem to Expand the Complex Number $( 5 + i ) ^ { 4 }$