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# Help, I don't know how to explane

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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 ?

Aug 13, 2019

#1
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$$(a+b)^n = \sum \limits_{k=0}^n \dbinom{n}{k}a^k b^{n-k}$$

$$\left(x^2 + \dfrac 1 4 \right)^4 = \sum \limits_{k=0}^4 \dbinom{4}{k}(x^2)^k \left(\dfrac 1 4\right)^{n-k}\\ \text{The constant term appears when k=0}\\ \dbinom{4}{0}\left(\dfrac 1 4\right)^4 = \dfrac{1}{256}$$

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Aug 14, 2019
#3
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Your expansion is correct! But you don't have any term that is independent of x, so the constant term is 0.

Aug 14, 2019
#4
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This isn't correct.  The constant term is 1/256 as shown above.

WA to verify this

https://www.wolframalpha.com/input/?i=(x%5E2+%2B+1%2F4)%5E4

Rom  Aug 14, 2019
#5
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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.

Alan  Aug 15, 2019