Use the Binomial Theorem to expand and simplify the expression. $\left( u ^ { 3 / 5 } + 5 \right) ^ { 5 }$
A) $u ^ { 3 } + 25 u ^ { 12 / 5 } - 1250 u ^ { 9 / 5 } - 250 u ^ { 6 / 5 } + 3125 u ^ { 3 / 5 } + 3125$
B) $u ^ { 3 } + 250 u ^ { 12 / 5 } + 25 u ^ { 9 / 5 } + 1250 u ^ { 6 / 5 } + 3125 u ^ { 3 / 5 } + 3125$
C) $u ^ { 3 } + 250 u ^ { 12 / 5 } + 1250 u ^ { 9 / 5 } + 3125 u ^ { 6 / 5 } + 3125 u ^ { 3 / 5 } + 3125$
D) $u ^ { 3 } + 25 u ^ { 12 / 5 } + 3125 u ^ { 9 / 5 } + 250 u ^ { 6 / 5 } + 3125 u ^ { 3 / 5 } + 1250$
E) $u ^ { 3 } + 25 u ^ { 12 / 5 } + 250 u ^ { 9 / 5 } + 1250 u ^ { 6 / 5 } + 3125 u ^ { 3 / 5 } + 3125$

Question

Use the Binomial Theorem to expand and simplify the expression.   $\left( u ^ { 3 / 5 } + 5 \right) ^ { 5 }$  
A) $u ^ { 3 } + 25 u ^ { 12 / 5 } - 1250 u ^ { 9 / 5 } - 250 u ^ { 6 / 5 } + 3125 u ^ { 3 / 5 } + 3125$
B) $u ^ { 3 } + 250 u ^ { 12 / 5 } + 25 u ^ { 9 / 5 } + 1250 u ^ { 6 / 5 } + 3125 u ^ { 3 / 5 } + 3125$
C) $u ^ { 3 } + 250 u ^ { 12 / 5 } + 1250 u ^ { 9 / 5 } + 3125 u ^ { 6 / 5 } + 3125 u ^ { 3 / 5 } + 3125$
D) $u ^ { 3 } + 25 u ^ { 12 / 5 } + 3125 u ^ { 9 / 5 } + 250 u ^ { 6 / 5 } + 3125 u ^ { 3 / 5 } + 1250$
E) $u ^ { 3 } + 25 u ^ { 12 / 5 } + 250 u ^ { 9 / 5 } + 1250 u ^ { 6 / 5 } + 3125 u ^ { 3 / 5 } + 3125$

Quizplus · Accepted Answer

The correct expansion using the Binomial Theorem involves calculating each term by combining coefficients and powers correctly. The pattern of coefficients follows Pascal's triangle, and for a binomial $(a + b)^n$, the $k$th term is given by $\binom{n}{k} a^{n-k}b^k$. For $(u^{3/5} + 5)^5$, the coefficients from Pascal's triangle for $n=5$ are 1, 5, 10, 10, 5, 1. Applying these to the terms gives the correct expansion as in option E.

Use the Binomial Theorem to Expand and Simplify the Expression (u3/5+5)5\left( u ^ { 3 / 5 } + 5 \right) ^ { 5 }(u3/5+5)5

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