Proofs of exponential rules, radical rules, and rational exponent rules
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blamocur
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I copy-pasted the title of your thread into Google, and two hits out of the first three seem useful -- did you see them?
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Proving the exponential laws for rational exponents
I have managed to prove the usual exponent laws for integer exponents, using induction on the natural numbers. I am now trying to prove that the laws also hold for rational exponents. I'm startin...
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Googled for 45 minutes and didn't see those articles. Thank you both for that. The andrusia article was very good. Very helpful for exponents and rational exponents, but still haven't found anything great for radicals/roots other than saying just to convert them to exponents.
topsquark
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I'll show you for a square root. The other radicals are similar.
I'm going to assume we have already proven that [imath]x^a \cdot x^b = x^{a + b}[/imath] for rational a and b
We know that the square function and the square root function are inverses of each other. So [imath]\left ( \sqrt{x} \right )^2 = x[/imath]. If the radical can be expressed as an exponent, then write it as [imath]\sqrt{x} = x^a[/imath].
So
[imath]\left ( \sqrt{x} \right )^2 = ( x^a \cdot x^a ) = x^{2a} = x = x^1[/imath]
But that means 2a = 1. Thus a = 1/2 and therefore [imath]\sqrt{x} = x^a = x^{1/2}[/imath]. Other radicals work the same way. [imath]( \sqrt[3]{x} )^3 = x[/imath], etc.
-Dan
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