Reason for radicals vs. rational exponents?

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jpanknin

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I'm revisiting math and trying to get a more intuitive feel rather than just memorize formulas and concepts. One that confuses me is why are radical symbols ( [imath]\sqrt{}[/imath] )used rather than rational exponents (i.e., [math]8^{\frac{1}{3}}=2[/math])? Everything I've come across on working with radicals says to convert them to rational exponents and then solve. So...why are radical symbols still used at all? Is there a good reason for using them?
 
jpanknin said:
I'm revisiting math and trying to get a more intuitive feel rather than just memorize formulas and concepts. One that confuses me is why are radical symbols ( [imath]\sqrt{}[/imath] )used rather than rational exponents (i.e., [math]8^{\frac{1}{3}}=2[/math])? Everything I've come across on working with radicals says to convert them to rational exponents and then solve. So...why are radical symbols still used at all? Is there a good reason for using them?
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There are several situations like this where we have a couple synonymous notations, like fractions and decimals.

I would say that radicals are, first, much easier to write for something basic like [imath]\sqrt{2}[/imath]. That alone is enough reason for radicals to still be used.

I think they also make some complicated expressions easier to follow, like maybe [imath]\frac{\sqrt[3]{x}-\sqrt[3]{y}}{\sqrt[3]{x}+\sqrt[3]{y}}[/imath]. And sometimes we just use radicals in one expression for consistency because we've used them in another.

To be honest, though, given [imath]8^{\frac{1}{3}}[/imath], I would first read it, if not write it, as [imath]\sqrt[3]{8}[/imath] in order to evaluate it!
 
Thanks, @Dr.Peterson. That makes sense. However, are there situations where leaving it in radical form (other than simple expressions like [imath]\sqrt[3]{8}[/imath]) would be easier/better to work with (solve) or do rational exponents make that easier?
 
jpanknin said:
Thanks, @Dr.Peterson. That makes sense. However, are there situations where leaving it in radical form (other than simple expressions like [imath]\sqrt[3]{8}[/imath]) would be easier/better to work with (solve) or do rational exponents make that easier?
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It may be partly a matter of familiarity (because we learn the radical form first), but it feels easier to simplify something like
[math]\sqrt[3]{81x^4y^5}=\sqrt[3]{27x^3y^3}\sqrt[3]{3x^1y^2}=3xy\sqrt[3]{3x^1y^2}[/math]than
[math](81x^4y^5)^\frac{1}{3}=(3^4)^\frac{1}{3}x^\frac{4}{3}y^\frac{5}{3}=3^{1+\frac{1}{3}}x^{1+\frac{1}{3}}y^{1+\frac{2}{3}}=3\cdot3^\frac{1}{3}x\cdot x^\frac{1}{3}y\cdot y^\frac{2}{3}=3xy(3xy^2)^\frac{1}{3}[/math]
 
Dr.Peterson said:
It may be partly a matter of familiarity (because we learn the radical form first), but it feels easier to simplify something like
[math]\sqrt[3]{81x^4y^5}=\sqrt[3]{27x^3y^3}\sqrt[3]{3x^1y^2}=3xy\sqrt[3]{3x^1y^2}[/math]than
[math](81x^4y^5)^\frac{1}{3}=(3^4)^\frac{1}{3}x^\frac{4}{3}y^\frac{5}{3}=3^{1+\frac{1}{3}}x^{1+\frac{1}{3}}y^{1+\frac{2}{3}}=3\cdot3^\frac{1}{3}x\cdot x^\frac{1}{3}y\cdot y^\frac{2}{3}=3xy(3xy^2)^\frac{1}{3}[/math]
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And, just to be my difficult self, I disagree. I feel that the second expression is easier to work with. However that might simply be how I was introduced to the topic in High School... we tended to use powers rather than radicals for anything that wasn't a square root.

-Dan
 
Since we are being difficult, I will more difficult that topsquark. I disagree with both of you. It is a matter of style/taste whether you write an expression using radicals or exponents. Sometimes you might think radicals is a better/nicer form and other times you might choose exponents.
All kidding aside, I truly believe that the answer to which you should use is entirely up to you. It is purely a matter of what you like.
 
jpanknin said:
I'm revisiting math and trying to get a more intuitive feel rather than just memorize formulas and concepts. One that confuses me is why are radical symbols ( [imath]\sqrt{}[/imath] )used rather than rational exponents (i.e., [math]8^{\frac{1}{3}}=2[/math])? Everything I've come across on working with radicals says to convert them to rational exponents and then solve. So...why are radical symbols still used at all? Is there a good reason for using them?
Click to expand...
I like "exponents" - it can be carried over to "complex analyses" very conveniently.
 
Very interesting thread. Thanks @Dr.Peterson, @topsquark, @Steven G, and @Subhotosh Khan. I've never seen each method side-by-side, but I can see situations that seem easier (to me) with both approaches. Appreciate the debate. It really was helpful to hear both sides and see how each approach would be simplified.
 
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