How do you simplify irrational exponents?

[ISX]

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 can't 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?

 

[TylerH]

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

 

[Mentallic]

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 can't 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 10^0.5 should be somewhere between 10⁰ and 10¹.

If you use the rule that \left(a^b\right)^c=a^{bc} then you can get a lot of these rules. For example, \left(10^{0.5}\right)^2=10^{0.5\cdot 2}=10^1=10 So that means 10^{0.5} 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 a^b\cdot a^c=a^{b+c}.