How do you simplify irrational exponents?

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It is clear that 10^2 can be simplified to 10*10=100. But what about say, 10^0.5? I have been thinking about this for days and cant figure out how it simplifies. 10^1 is 10, 10^0 is 1, so 10^0.5 should be under 1, but it is 3.16, so I don't get it. Same with 10^-1 is 0.1. How exactly are those simplified?
 

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They aren't. 10^.5 = sqrt(10). Square roots of non-perfect squares are irrational, so you can't write it any other way than as a square root or rational exponent(ie to the power of 1/2).

Any number to a negative power is equal to the number to the power under 1. Ex. 10^-x = 1/10^x
 
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It is clear that 10^2 can be simplified to 10*10=100. But what about say, 10^0.5? I have been thinking about this for days and cant figure out how it simplifies. 10^1 is 10, 10^0 is 1, so 10^0.5 should be under 1, but it is 3.16, so I don't get it. Same with 10^-1 is 0.1. How exactly are those simplified?
If 101=10, 100=1, then 100.5 should be somewhere between 100 and 101.

If you use the rule that [tex]\left(a^b\right)^c=a^{bc}[/tex] then you can get a lot of these rules. For example, [tex]\left(10^{0.5}\right)^2=10^{0.5\cdot 2}=10^1=10[/tex] So that means [tex]10^{0.5}[/tex] is whatever number that when you multiply it by itself (square it) you get 10. This is the square root of 10.

You can use a similar idea to find out what negative exponents do. Use the fact that [tex]a^b\cdot a^c=a^{b+c}[/tex].
 

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