Proof of x^{3/2} = \sqrt{x^{3}} is a typo?

In summary, something to note is that for any positive integers m and n, (a^m)^n = a^{mn}. This can be easily shown by expanding the left-hand side and simplifying. However, this proof only works for positive integer exponents. For non-integer exponents, we must use the properties of exponents to define a^x. This can be done by considering the continuity of the function f(x)=a^x.
  • #1
Hootenanny
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Something I have been curious about, but never had the time to think about is, I know that;

[tex]x^{3/2} = \sqrt{x^{3}}[/tex]

But I have never seen any proof of this. Does anyone have a good resource or can show me the proof here? It would be much appreciated.

~H
 
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  • #2
So are you asking why [tex](a^m)^n=a^{mn}[/tex]?

A simple explanation would just be to expand the left-hand side:

[tex](a^m)^n[/tex]
[tex]=a^m \cdot a^m \cdot...\cdot a^m[/tex] (n-times)
[tex]=a^{m+m+m+...+m}[/tex] (n times) since [itex]a^m \cdot a^m=a^{m+m}[/itex]
[tex]=a^{mn}[/tex] since m+m+m+...+m n times is just m times n
 
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  • #3
dav2008 said:
So are you asking why [tex](a^m)^n=a^{mn}[/tex]?

A simple explanation would just be to expand the left-hand side:

[tex](a^m)^n[/tex]
[tex]a^m \cdot a^m \cdot...\cdot a^m[/tex] (n-times)
[tex]a^{m+m+m+...+m}[/tex] (n times) since [itex]a^m \cdot a^m=a^{m+m}[/itex]
[tex]a^{mn}[/tex] since m+m+m+...+m n times is just m times n

Thank's yeah, I've just got it. Just as I was replying to this I found it in one of my old textbooks, guess I should look through my books more before asking stupid questions.:grumpy: Thank's again.

~H
 
  • #4
I guess that proof only works for (positive) integer exponents.

I googled "properties of exponents proof" and found this webpage: http://planetmath.org/encyclopedia/ProofOfPropertiesOfTheExponential.html

That might explain it some more for non-integer exponents.
 
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  • #5
This is not so much a "proof" as an explanation of why we define
ax in certain ways.

It is easy to show that, as long as n and m are positive integers, anam= an+m. That's just a matter of counting the number of "a"s being multiplied.
Similarly, it is easy to show that, as long as n and m are positive integers, (an)[/sup]m[/sup]= anm. Again, that's just a matter of counting the number of "a"s being multiplied.

Those are very nice formulas! It would help a lot if axay= ax+y and (ax)y= axy for all x and y.

IF it were true that ana0= an+0, even when the exponent is 0, we must have an+0= an= ana0 and if a is not 0 we can divide by an to get a0= 1 as long as a is not 0.

Similarly, to guarantee that this formula is true for n negative, we must have ana-n= an-n= a0= 1: in other words that a-n= 1/an. Of course, we can only do that division if an is not 0: in other words if a is not 0.

If we want (an)m even when m is not a positive, we must have (an)-n= an-n= a0= 1. In other words, we must define a0= 1 as long as a is not 0. There is no way to define 00 that will make anam= an+m for a= 0.

Similarly, if we want anam= an+m for n or m negative, we must have ana-n= an-n= a0= 1 for all positive integers n: in other words, again dividing by an, a-n= 1/an as long as a is not 0.

As dav2008 pointed out, in order to have (am)n= amn true even when m and n are not integers, we must have (an)1/n= a1 so that a1/n= [itex]\^n\sqrt{a}[/itex] and then, that am/n= [itex]^n\sqrt{a^n[/itex].

In order to define ax for x irrational we require that f(x)= ax be continuous.
 
  • #6
Ahh, makes more sense now. Thanks both of you, it is much appreciated.

~H
 
  • #7
HallsofIvy said:
If we want (an)m even when m is not a positive, we must have (an)-n= an-n= a0= 1.
One of us is making a mistake with this bit. Is that a typo ?
 
  • #8
Gokul43201 said:
One of us is making a mistake with this bit. Is that a typo ?

Didn't spot that, surely (an)-n = [itex]a^{-n\cdot n}[/itex]?

~H
 

1. What is the purpose of an indices proof?

An indices proof is used to show that a mathematical statement involving exponents, or indices, is true. This is often done by manipulating the indices according to the rules of exponents.

2. How do I know if an indices proof is correct?

An indices proof is considered correct if each step follows logically from the previous step and if the final statement is equivalent to the original statement. In other words, the proof should be a series of valid mathematical deductions that result in the desired statement.

3. What are some common techniques used in indices proofs?

Some common techniques used in indices proofs include using the laws of exponents, simplifying expressions using properties of exponents, and using algebraic manipulation to transform the original statement into an equivalent one.

4. Can indices proofs be used for any mathematical statement involving exponents?

Yes, indices proofs can be used for any mathematical statement involving exponents, as long as the statement can be rewritten using the rules of exponents and the proof follows logical steps.

5. Are there any tips for successfully completing an indices proof?

Some tips for successfully completing an indices proof include carefully reading the given statement, clearly stating the steps in the proof, and checking each step for accuracy. It can also be helpful to start by simplifying the expression as much as possible before attempting the proof.

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