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I have come across this question and I cannot understand the step highlighted. I would have expected that the fractional exponents of the terms in the numerator would have a negative value after multiplying by the term in the denominator. Can someone be kind enough to clarify.

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3 Answers

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$$\frac{{x - \sqrt x }}{{{x^{\frac{1}{3}}}}} = \frac{{{x^1}}}{{{x^{\frac{1}{3}}}}} - \frac{{{x^{\frac{1}{2}}}}}{{{x^{\frac{1}{3}}}}} = {x^{1 - \frac{1}{3}}} - {x^{\frac{1}{2} - \frac{1}{3}}} = {x^{\frac{2}{3}}} - {x^{\frac{1}{6}}}$$

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When dividing, the exponents subtract. So $$\frac{x - \sqrt{x}}{x^{1/3}} = \frac{x^1}{x^{1/3}} - \frac{x^{1/2}}{x^{1/3}} = x^{1 - 1/3} - x^{1/2 - 1/3} = x^{2/3} - x^{1/6}$$

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What is going here is that when exponents are divided, they subtract each other.

So:

$$\large\frac{x - \sqrt{x}}{x^{1/3}}$$ $$= \large x^{1-1/3} - x^{1/2-1/3}$$ $$= \large x^{2/3} - x^{1/6}$$

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