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 Binomial Theorem states that $(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$. Applying this to $(u^{3/5} + 5)^5$, we calculate each term by choosing $u^{3/5}$ and $5$ appropriately for each $k$ from 0 to 5. The coefficients are determined by $\binom{5}{k}$ and the powers of $u$ and $5$ are adjusted accordingly. The correct expansion and simplification lead to the expression in choice 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 $\left( u ^ { 3 / 5 } + 5 \right) ^ { 5 }$