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ScienceNostalgia101

Dead end in radical simplification?

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So I'm reviewing my rules of radicals prior to teaching it to students, and found out I'm a little rusty on them.

Suppose you hit an answer that ends with a prime number as your radicand. Provided you used mathematically valid reasoning to get there, does this prime-number radicand now suggest that you arrived at the most simplified form, or are there "dead ends" distinct from the right answer?

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Posted (edited)

It doesn't have to be prime- as long as a number is not a perfect square or divisible by a perfect square,  you cannot continue.  For example to simplify the square root of 216, I can observe 216= 2*2*2*3*3*3= 2^3 3^3 (that's its "prime factorization").   Since I want the square root, I look for squares- powers of 2: (2*2)(2)(3*3)(3)= 4(2)(9)(3). "4" and "9" are "perfect squares", 2 squared and 3 squared.  [math]\sqrt{216}= \sqrt{4(9)(2)(3)= \sqrt{4}\sqt{9}\sqrt{2(3)}= 2(3)\sqrt{6}= 6\sqrt{6}[/math].  "6" is not  prime but it is not a perfect square either.  

Notice that 2*2*2= 8 and 3*3*3= 27 are "perfect cubes", $2^3$ and $3^3$ so its cube root: [math]\sqrt[3]{216}=  \sqrt[3]{2^3(3^3}= 2(3)= 6[/math].

Edited by Country Boy

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