Use the Binomial Theorem to expand and simplify the expression. $\left( x ^ { 3 / 4 } + 5 \right) ^ { 4 }$
A) $x ^ { 3 } + 625$
B) $x ^ { 3 } + 20 x ^ { 9 / 4 } + 150 x + 500 x ^ { 3 / 4 } + 625$
C) $x ^ { 3 } + 20 x ^ { 9 / 5 } + 150 x ^ { 3 / 2 } + 500 x ^ { 3 / 4 } + 625$
D) $x ^ { 3 } + 20 x ^ { 9 / 4 } + 150 x ^ { 3 / 2 } + 500 x ^ { 3 / 4 } + 625$
E) $- x ^ { 3 } + 20 x ^ { 9 / 4 } + 150 x ^ { 3 / 2 } + 500 x + 625$

Question

Quizplus · Accepted Answer

Using the Binomial Theorem, we have:$\left( x ^ { 3 / 4 } + 5 \right) ^ { 4 } = \sum_{k=0}^{4} \binom{4}{k} \left( x ^ { 3 / 4 } \right) ^ { 4 - k } 5 ^ { k }$$\Rightarrow \left( x ^ { 3 / 4 } + 5 \right) ^ { 4 } = \binom{4}{0} \left( x ^ { 3 / 4 } \right) ^ { 4 } + \binom{4}{1} \left( x ^ { 3 / 4 } \right) ^ { 3 } 5 ^ { 1 } + \binom{4}{2} \left( x ^ { 3 / 4 } \right) ^ { 2 } 5 ^ { 2 } + \binom{4}{3} \left( x ^ { 3 / 4 } \right) ^ { 1 } 5 ^ { 3 } + \binom{4}{4} \left( x ^ { 3 / 4 } \right) ^ { 0 } 5 ^ { 4 }$$\Rightarrow \left( x ^ { 3 / 4 } + 5 \right) ^ { 4 } = 1 \cdot x ^ { 3 } + 4 \cdot x ^ { 9 / 4 } \cdot 5 + 6 \cdot x ^ { 3 / 2 } \cdot 5 ^ { 2 } + 4 \cdot x ^ { 3 / 4 } \cdot 5 ^ { 3 } + 1 \cdot 5 ^ { 4 }$$\Rightarrow \left( x ^ { 3 / 4 } + 5 \right) ^ { 4 } = x ^ { 3 } + 20 x ^ { 9 / 4 } + 150 x ^ { 3 / 2 } + 500 x ^ { 3 / 4 } + 625$

Use the Binomial Theorem to Expand and Simplify the Expression (x3/4+5)4\left( x ^ { 3 / 4 } + 5 \right) ^ { 4 }(x3/4+5)4

Use the Binomial Theorem to Expand and Simplify the Expression $\left( x ^ { 3 / 4 } + 5 \right) ^ { 4 }$