Suggestions on how to go about proving a^(m+n)=a^(m)a^(n)

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In summary, by using the properties of exponents and the principle of induction, it can be proven that for any nonzero number a and integers m and n, the equality a^(m+n) = a^m * a^n holds true. This can be shown for both positive and negative values of m and n.
  • #1
alianna
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Let a be a nonzero number and m and n be integers. Prove the following equality. a^(m+n)=a^(m)a^(n)


Im not really sure what direction to go in. I am not sure if I need to show for n positive and negative separately or is there an easier way.


My attempt/ideas:
When n>0: a^(m)a^(n)= (a*a*...*a)(m times) *(a*a*a*a*...*a)(n times)
=a*a*...*a(m+n times)
=a^(m+n)
When n<0: a^(m)a^(n)=a^(m)=a...a(m times)/ a...a(n times)
 
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  • #2
Are you familiar with induction?
 
  • #3
Yes we just went over it but i wasn't sure how to go about induction with m and n being integers...
 
  • #4
1) Take ##m=0##. Prove that ##a^{m + n} = a^m a^n##. This shouldn't be a problem.

2) Assume the result holds for ##m##. Prove it holds for ##m+1##. So you know that ##a^{m+n} = a^m a^n##. You need to prove ##a^{m + n + 1} = a^{m+1} a^n##.
 
  • #5
So would you be able to say: Since a^(m+n)=a^(m)a^(n).
Then this implies that: a^(m+n+1)= a^(m+1+n)= a^(m+1)a^(n)?

This doesn't seem right because we are assuming what we are trying to prove.
Why are you allowed to show for m+1 but don't have to for n+1 or do you just assume since m+1 works then n+1 works?
Im still confused how this deals with the negative values of m and n.
 
  • #6
alianna said:
So would you be able to say: Since a^(m+n)=a^(m)a^(n).
Then this implies that: a^(m+n+1)= a^(m+1+n)= a^(m+1)a^(n)?

This doesn't seem right because we are assuming what we are trying to prove.
Why are you allowed to show for m+1 but don't have to for n+1 or do you just assume since m+1 works then n+1 works?
Im still confused how this deals with the negative values of m and n.

Hint :

##a^{m+n+1} = aaaa...a##

(m+n) + 1 times.
 

What is the equation for proving a^(m+n)=a^(m)a^(n)?

The equation is a^(m+n)=a^(m)a^(n), where "a" is the base and "m" and "n" are the exponents.

What is the first step in proving a^(m+n)=a^(m)a^(n)?

The first step is to rewrite the equation in exponential form, which is a^(m+n)=(a^m)(a^n).

What is the second step in proving a^(m+n)=a^(m)a^(n)?

The second step is to use the properties of exponents to simplify the equation. In this case, we can use the product rule of exponents, which states that a^(m+n)=a^m*a^n.

Why is it important to show that a^(m+n)=a^(m)a^(n)?

It is important to show this equation because it is a fundamental property of exponents and is used in many mathematical proofs and calculations.

Can this equation be applied to all values of "a", "m", and "n"?

Yes, this equation can be applied to all real numbers for "a" and integers for "m" and "n". It is a general rule that holds true for all values of the variables.

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