Here is the question,
Find the constant term in the expansion of,
\(\Big(x^2+\frac{1}{x}\Big)^4\)
When I expand this equation I get,
\(\frac{\left(x^3+1\right)^4}{x^4}\)
I think this is the correct expantion but I also got another answer,
\(\frac{x^{12}+4x^9+6x^6+4x^3+1}{x^4}\)
I dont know what the constant term is and if there is not what should I put as the answer ?
Thx in advance.
+6248 \((a+b)^n = \sum \limits_{k=0}^n \dbinom{n}{k}a^k b^{n-k}\)
\(\left(x^2 + \dfrac 1 4 \right)^4 = \sum \limits_{k=0}^4 \dbinom{4}{k}(x^2)^k \left(\dfrac 1 4\right)^{n-k}\\ \text{The constant term appears when $k=0$}\\ \dbinom{4}{0}\left(\dfrac 1 4\right)^4 = \dfrac{1}{256} \)
.
+9519 Your expansion is correct! But you don't have any term that is independent of x, so the constant term is 0.
+6248 This isn't correct. The constant term is 1/256 as shown above.
WA to verify this
+33614 Rom has expanded \((x^2+\frac{1}{4})^4\) but the expression in the original question is \((x^2+\frac{1}{x})^4\)
Given that the question asks for the constant term, Rom's version might make more sense; however, it is different from the original, for which Max's answer is correct.
