Find the constant term in the expansion of...

Question

How can we find the term independent of $x$ in the expansion of,

$(x^{\frac{2}{3}}+4x^{\frac{1}{3}}+4)^5\cdot\left(\dfrac{1}{x^{\frac{1}{3}}-1}+\dfrac{1}{x^{\frac{2}{3}}+x^{\frac{1}{3}}+1}\right)^{-9}$

Thank you in advance !

Answer 1

Hint: The expression is equal to (use some algebraic identities): $$\frac{(x^{1/3}+2)(x-1)^9}{x^3}=2\binom93(-1)^6+\text{other terms with non-zero powers of x}$$

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Further Hint: $$(x-1)(x^2+x+1)=(x^3-1)$$

Answer 2

Hint

$$\dfrac{1}{x^{\frac{1}{3}}-1}+\dfrac{1}{x^{\frac{2}{3}}+x^{\frac{1}{3}}+1}=\frac{\left(\sqrt[3]{x}+2\right) \sqrt[3]{x}}{x-1}$$ $$x^{\frac{2}{3}}+4x^{\frac{1}{3}}+4=\left(\sqrt[3]{x}+2\right)^2$$ $$(x^{\frac{2}{3}}+4x^{\frac{1}{3}}+4)^5\cdot\left(\dfrac{1}{x^{\frac{1}{3}}-1}+\dfrac{1}{x^{\frac{2}{3}}+x^{\frac{1}{3}}+1}\right)^{-9}=\frac{\left(\sqrt[3]{x}+2\right) (x-1)^9}{x^3}$$

I am sure that you can take from here.

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