Find the 5^("th ") term in the expansion of (x+3y)^(5) in simplest form. Answer:

Identify General Term: Identify the general term of the binomial expansion. The general term in the expansion of (a+b)^n is given by T(k+1) = C(n, k) * a^(n-k) * b^k, where C(n, k) is the binomial coefficient ＂n choose k＂.Determine Specific Term: Determine the specific term we are looking for. We want to find the 5th term, which corresponds to k=4 in the general term formula, since the first term corresponds to k=0.Calculate Binomial Coefficient: Calculate the binomial coefficient for the 5th term. C(5, 4) = 5! / (4!(5-4)!) = 5 / 1 = 5.Substitute Values: Substitute the values into the general term formula to find the 5th term. T(5) = C(5, 4) * x^(5-4) * (3y)^4 = 5 * x^1 * (3y)^4.Simplify Term: Simplify the 5th term. T(5) = 5 * x * (3y)^4 = 5 * x * (81y^4) = 405x * y^4.

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Find the
5^("th ") term in the expansion of
(x+3y)^(5) in simplest form.
Answer:

Find the $5^{\text {th }}$ term in the expansion of $(x+3 y)^{5}$ in simplest form. $\newline$ Answer:

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Math Problems

Algebra 1

Identify a sequence as explicit or recursive

Full solution

Q. Find the

5^{\text {th }}

term in the expansion of

(x+3 y)^{5}

in simplest form.

\newline

Answer:

Identify General Term: Identify the general term of the binomial expansion. $\newline$ The general term in the expansion of $(a+b)^n$ is given by $T(k+1) = C(n, k) \cdot a^{(n-k)} \cdot b^k$ , where $C(n, k)$ is the binomial coefficient ＂n choose k＂.
Determine Specific Term: Determine the specific term we are looking for. $\newline$ We want to find the $5^{\text{th}}$ term, which corresponds to $k=4$ in the general term formula, since the first term corresponds to $k=0$ .
Calculate Binomial Coefficient: Calculate the binomial coefficient for the $5^{\text{th}}$ term. $C(5, 4) = \frac{5!}{4!(5-4)!} = \frac{5}{1} = 5$ .
Substitute Values: Substitute the values into the general term formula to find the $5^{\text{th}}$ term. $T(5) = C(5, 4) \times x^{(5-4)} \times (3y)^4 = 5 \times x^1 \times (3y)^4.$
Simplify Term: Simplify the $5^{\text{th}}$ term. $T(5) = 5 \times x \times (3y)^4 = 5 \times x \times (81y^4) = 405x \times y^4$ .