Fractional Exponents (How is it done?)

[Bored]

How does 2^5/2 become 2^2 multiplied by 2^1/2?
(The '^' means 'to the power of' so 2 to the power of 5/2. I am not sure how to write this as an exponent as this is my first post.)

2^5/2 = 2^2 × 2^1/2

So 2^2 = 4 and 2^1/2 means Square Root so there is a radical sign, so it becomes √2.

I tried to reverse engineer the solution but I'm still not sure how 2^5/2 makes it. I know you first Square root it so √2 then you put to the power of 5 so √2^5. This goes into decimals so I am confused on what to do next. I looked at the solution and I can see that both of them make 2^5/2 I just don't know how it was factorized into those specific numbers, because 4 x √2 = 4√2 and somehow that is √2^5? I know 2^5 also makes 32.

So I think I might of done something wrong or I don't know the correct method, can someone please attempt to help me and thank you for your time!

 

[johnqwertyful]

Do you agree that x^(a+b)=(x^a) (x^b) for natural x,a,b?

edit, forgot a +

 

[Bored]

In my case would that be 2^5/2 (5 is a and 2 is b) = (2^5) (5^2)? I'm quite sure I'm doing something wrong so I think I can't agree on that even though I think its correct.

So what is a and b? If a and b are 5 and 2 they make (2^5) (5^2) making 800. I am kind of lost... Am I getting it wrong?

Another go at it I see that 2^5 = 32 and 5^2 = 25 - So do I put in the fraction making it 2^5/1 and 5^1/2?

Oh now I see it and the '+', Thank you for changing it

 

[johnqwertyful]

Do you agree that
2^4 = 2^2 * 2^2
3^{10}=3^4 * 3^6

 

[Bored]

Yes, Because when you are multiplying powers you add them together, but since I started fractional exponents I am very confused - probably because of the fractions.

So if you added 2^2 and 2^1/2 that would make 2^3/2 if i am correct or 2.5?

Okay so there's 2/1 + 1/2 = 2 5/2
So adding fractions you double it to make it 4/2 then add it to make it 5/2 making it 2^5/2.
So we done this backwards but how did 2^5/2 split up to make those 2?

Oh wait (Edit), So you could also spilt it up to something else? But it has to be a Fractional Exponents because so it can be combined? Not sure why it has to be a fractional exponent but I know you split the expression into the product of an integer exponent and the factor with a fractional coefficient less than one. I think the reason why this is done is so you can keep the root in the equation, otherwise it no longer has a fractional exponent.

Thank you for your valuable time, which you were very helpful!

PS. If I made any mistakes please correct (:

 

[johnqwertyful]

2+\frac{1}{2}=\frac{4}{2}+\frac{1}{2}=\frac{5}{2}

 

[mfb]

2^5/2 is problematic, it could mean $\frac{2^5}{2}$. You can write it as 2^(5/2) or better [noparse]2^5/2[/noparse], which gets parsed as 2^5/2. And you can use LaTeX here.

You can split up the 5/2 like this:
$$\frac{5}{2}=\frac{4+1}{2}=\frac{4}{2}+\frac{1}{2}=2+\frac{1}{2}$$
And then use the rule for exponents written above.