<s>[INST] radicals and rational exponents. simplifying $4^{5/3}$ This question is not for homework.
The book says:
$$4^{5/3} = (\sqrt4)^5 = (\sqrt{2^2})^5 = 2^5 = 32$$
when a bunch of us tried it:
$$4^{5/3} = \sqrt[3]{4^5} = \sqrt[3]{2^3 \cdot 2^2} = 2 \cdot \sqrt[3]{2^2} = 2 \cdot \sqrt[3]{4}$$
I get the sense we missed something. Where did we mess it up? We don't know how to get to 32.
 [/INST] There is probably a typo in your book, since
$$(\sqrt 4)^5 = 4^{5/2}\neq 4^{5/3}.$$
However, you still made a mistake in your calculation. You replaced $4^5$ with $2^3\cdot 2^2$, when in reality, $4^5 = 4^3\cdot 4^2 = 2^6 \cdot 2^4$.
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