Unformatted text preview: a 2 / 3 to be the number 3 √ a 2 . Prove that this number is equal to ( 3 √ a ) 2 . (Hint: By the Cubic Root Theorem, this number is the unique solution to the equation x 3 = a 2 .) Let p,q to be positive integers such that p/q = 2 / 3. Deﬁne the symbol a p/q to be the number q √ a p . Prove that this number is also equal to 3 √ a 2 . 6. The Fractional Power Law. Do this one for fun. Do not hand in your proof but feel free to discuss it with me. Let p,q,r,s be positive integers. For positive a ∈ R , we deﬁne the symbol a p/q to be the number q √ a p . Prove that a p/q a r/s = a p/q + r/s and that ( a p/q ) r/s = a ( pr ) / ( qs ) . (Hint: Consider the equations x qs = a ps + qr and x qs = a pr respectively and use the n th Root Theorem.) You can also prove that if p/q = r/s , then q √ a p = s √ a r by a similar consideration....
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- Spring '09
- Math, positive integers, Nth root, root theorem