Find the 4 th term in the expansion of (x+2y)^(6) in simplest form. Answer:

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Find the 4 th term in the expansion of  (x+2y)^(6) in simplest form. Answer:

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Apply Binomial Theorem: To find the 4th term in the expansion of (x+2y)^(6), we will use the binomial theorem. The binomial theorem states that the nth term in the expansion of (a+b)^(m) is given by the formula C(m, k) * a^(m-k) * b^k, where C(m, k) is the binomial coefficient ＂m choose k＂. For the 4th term, k will be 3 because the first term corresponds to k=0.Calculate Binomial Coefficient: We calculate the binomial coefficient for the 4th term, which is C(6, 3). This is equal to 6! / (3! * (6-3)!), where ＂!＂ denotes factorial.Simplify Factorials: Calculating the factorials, we get 6! = 6 * 5 * 4 * 3 * 2 * 1, 3! = 3 * 2 * 1, and (6-3)! = 3! = 3 * 2 * 1. Substituting these into the binomial coefficient formula, we have C(6, 3) = (6 * 5 * 4) / (3 * 2 * 1).Calculate Binomial Coefficient: Simplifying the binomial coefficient, we get C(6, 3) = (6 * 5 * 4) / (3 * 2 * 1) = 20.Calculate Powers: Now we need to calculate the rest of the term, which is x^(6-3) * (2y)^3. This simplifies to x^3 * (2y)^3.Simplify Powers: Calculating the powers, we get x^3 * (2y)^3 = x^3 * 8y^3, since (2y)^3 = 2^3 * y^3 = 8y^3.Multiply Coefficients and Powers: Multiplying the binomial coefficient by the calculated powers, we get the 4th term: 20 * x^3 * 8y^3.Simplify Final Term: Simplifying the expression, we get the 4th term in simplest form: 160x^3y^3.

Find the $4$ th term in the expansion of $(x+2 y)^{6}$ in simplest form. $\newline$ Answer:

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Find the $4$ th term in the expansion of $(x+2 y)^{6}$ in simplest form. $\newline$ Answer: