# Simplified form of $x^{10/3}$

I'm in a intermediate algebra class and am confused about how to get the simplified form of $\sqrt{x^{10}}$

I tend to want to write it as $x^{10/3}$ creating a mixed fraction then simplifying that to get $x^{3}\sqrt{x}$

However, when asking a friend they explained that if we look at it by going $\sqrt{x^{8}}\sqrt{x^{2}}$ then would get $x^2\sqrt{x^{2}}$

Could someone please help with which one is correct and if the top one is correct explain why.

• Please fix the title - it is a bit misleading. May 24, 2012 at 19:43
• We have $x^a\cdot x^b= x^{a+b}$ and $(x^a)^b=x^{ab}$. May 24, 2012 at 19:45
• @AD.feel free to change the title. I also understand exponent rules but for some reason the radical is throwing me off May 24, 2012 at 19:47
• BTW welcome to math.SE May 24, 2012 at 19:56
• "Simplify" is a term that cannot be defined precisely. A "simplification" that is best for one purpose is not necessarily best for another. I think that $x^{10/3}$ is a good general purpose simplification. But in a class, what is best is effectively what teacher thinks best. May 24, 2012 at 21:12

$x^2\sqrt{x^2}=x^2\cdot x^{2/3}=x^{8/3}\neq x^{10/3}$ ...
The mistake is at $\,\,\sqrt{x^8}\sqrt{x^2}\neq x^2\sqrt{x^2}\,\,$ since $\,\,\sqrt{x^8}\neq x^2\,\,$ but $\,x^{8/3}$
• so $x^{3}\sqrt{x}$ is correct? May 24, 2012 at 19:51
• Yes it is! $\,\,\,\,$ May 24, 2012 at 19:54
We have $$x^{10/3}=x^{(9 + 1)/3}=x^{9/3 + 1/3}=x^{3 + 1/3}=x^3x^{1/3}$$
you can try $x^6*x^4$,we in your case it would be $x^2*x^{4/3}$,there are many forms,for example as @AD. indicated