Question : The value of $(x^{\frac{1}{3}}+x^{-\frac{1}{3}})(x^{\frac{2}{3}}-1+x^{-\frac{2}{3}})$ is:
Option 1: $x^{-1}+x^{\frac{2}{3}}$
Option 2: $x+x^{-\frac{1}{3}}$
Option 3: $x^{\frac{1}{3}}+x^{–1}$
Option 4: $x+x^{–1}$
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Correct Answer: $x+x^{–1}$
Solution :
Given: $(x^{\frac{1}{3}}+x^{-\frac{1}{3}})(x^{\frac{2}{3}}-1+x^{-\frac{2}{3}})$
$=x^{\frac{1}{3}}×x^{\frac{2}{3}}-x^{\frac{1}{3}}+x^{\frac{1}{3}}×x^{-\frac{2}{3}}+x^{-\frac{1}{3}}×x^{\frac{2}{3}}-x^{-\frac{1}{3}}+x^{-\frac{1}{3}}×x^{-\frac{2}{3}}$
$=x^{\frac{1}{3}+\frac{2}{3}}-x^{\frac{1}{3}}+x^{\frac{1}{3}-\frac{2}{3}}+x^{-\frac{1}{3}+\frac{2}{3}}-x^{-\frac{1}{3}}+x^{-\frac{1}{3}-\frac{2}{3}}$
$=x^1-x^{\frac{1}{3}}+x^{-\frac{1}{3}}+x^{\frac{1}{3}}-x^{-\frac{1}{3}}+x^{-1}$
$=x+x^{-1}$
Hence, the correct answer is $x+x^{-1}$.
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