Apply the binomial theorem (a+b)^n = sum_k=0^n nk a^n-k b^k with a=x, b=2y, and n=5.

Question

Apply the binomial theorem (a+b)^n = sum_k=0^n nk a^n-k b^k with a=x, b=2y, and n=5.

Asked on March 29, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-03-29T10:44:12.196Z

Here are the solutions to the problems: 1. Find the binomial expansion of $(x+2y)^5$. Step 1: Apply the binomial theorem $(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$ with $a=x$, $b=2y$, and $n=5$. $$ (x+2y)^5 = \binom{5}{0} x^5 (2y)^0 + \binom{5}{1} x^4 (2y)^1 + \binom{5}{2} x^3 (2y)^2 + \binom{5}{3} x^2 (2y)^3 + \

Accepted Answer · 2026-03-29T10:44:12.196Z

Here are the solutions to the problems: 1. Find the binomial expansion of $(x+2y)^5$. Step 1: Apply the binomial theorem $(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$ with $a=x$, $b=2y$, and $n=5$. $$ (x+2y)^5 = \binom{5}{0} x^5 (2y)^0 + \binom{5}{1} x^4 (2y)^1 + \binom{5}{2} x^3 (2y)^2 + \binom{5}{3} x^2 (2y)^3 + \