Find the 6th term in the expansion of (x+2y)^(8) in simplest form. Answer:

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Find the 6th term in the expansion of  (x+2y)^(8) in simplest form. Answer:

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Use Binomial Theorem: To find the 6th term in the expansion of (x+2y)^(8), 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 6th term, k will be 5 because the first term corresponds to k=0.Calculate Binomial Coefficient: First, we need to calculate the binomial coefficient C(8, 5). This is equal to 8! / (5! * (8-5)!), which simplifies to 8! / (5! * 3!).Calculate Factorials: Calculating the factorial values, we get 8! = 8 * 7 * 6 * 5!, 5! = 5 * 4 * 3 * 2 * 1, and 3! = 3 * 2 * 1. Substituting these into the binomial coefficient formula, we have C(8, 5) = (8 * 7 * 6) / (3 * 2 * 1).Calculate Binomial Coefficient: Simplifying the fraction, we get C(8, 5) = (8 * 7 * 6) / (3 * 2) = 56.Calculate Remaining Term: Now we need to calculate the rest of the term, which is x^(8-5) * (2y)^5. This simplifies to x^3 * (2y)^5.Calculate Power of 2y: Raising 2y to the 5th power gives us 2^5 * y^5, which is 32 * y^5.Multiply Coefficients: Multiplying the binomial coefficient by the powers of x and y, we get the 6th term: 56 * x^3 * 32 * y^5.Calculate Final Term: Finally, we multiply 56 by 32 to get the coefficient of the 6th term. This gives us 56 * 32 = 1792.Final Simplified Term: The 6th term in the expansion of (x+2y)^(8) in simplest form is therefore 1792 * x^3 * y^5.

Find the $6$ th term in the expansion of $(x+2 y)^{8}$ in simplest form. $\newline$ Answer:

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Find the $6$ th term in the expansion of $(x+2 y)^{8}$ in simplest form. $\newline$ Answer: