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Following these threads: Thread 1 and Thread 2, I am still confused about the way we define functions with fractional exponents.

For example, $f(x)=(x^2)^\frac{3}{2}$. Is it true to say that $f(x)=x^3$ when $x\geq0$ and $f(x)=-x^3$ when $x<0$? In general, should we define $f(x)=(x^k)^\frac{a}{b} = (\sqrt[b]{x^k})^a $ or $f(x)=(x^k)^\frac{a}{b} = (\sqrt[b]{x^{ka}}) $?

Thank you!

Dr. John
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    "Is it true to say that $f(x) = x^3$ when $x \geq 0$ and $f(x) = -x^3$ when $x < 0$?" Yes, that is correct. – littleO Jun 16 '21 at 06:35

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