Can someone help me prove that $(a^m)^{1/n} = (a^{1/n})^m$ per the textbook excerpt captured in the image?
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1$(a^m)^{1/n} = (a^{m/n}) = (a^{1/n})^m$ – Adam May 26 '16 at 18:31
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For those landing on this page. The "proofs" here are at best incomplete, with gaps as large as the entirety of what is wanted to be proven, and make claims that are not true in general (they need restrictions on the values of the variables involved and these restrictions are not mentioned). In other words, assume everything in this page is wrong. Better take a look either here or here. – ameg May 31 '24 at 19:53
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Something like this?
$$(a^m)^{1/n} = a^{m \cdot (1/n)} = a^{(1/n) \cdot m} = (a^{1/n})^m$$
John
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Write down the definition:
$$\sqrt[n]{a^m}:=a^{m/n}\stackrel{\text{exponents prop.}}=\left(a^{1/n}\right)^m=\left(\sqrt[n]a\right)^m$$
DonAntonio
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