1. ## Exponents

Why is 5-⅔ (5 raised to the power of -2/3) equal to 1/5⅔ (1 divided 5 raised to the power of 2/3)? I mean to say how does the negative sign of the exponent ⅔ change to positive when 5-⅔ (5 raised to the power of -2/3) is inversed?

Thanks,

Ron

2. Originally Posted by rn5a
Why is 5-⅔ (5 raised to the power of -2/3) equal to 1/5⅔ (1 divided 5 raised to the power of 2/3)? I mean to say how does the negative sign of the exponent ⅔ change to positive when 5-⅔ (5 raised to the power of -2/3) is inversed?

Thanks,

Ron
Because $\displaystyle a^{-n} = \frac{1}{a^n}$.

3. Originally Posted by Prove It
Because $\displaystyle a^{-n} = \frac{1}{a^n}$.
But why is $\displaystyle a^{-n} = \frac{1}{a^n}$?

Ron

4. Originally Posted by rn5a
But why is $\displaystyle a^{-n} = \frac{1}{a^n}$?

Ron
Because it is. You can explain it using this formula :

$\displaystyle \frac{a^m}{a^n} = a^{m - n}$

In our case, $\displaystyle m = 0$ because $\displaystyle a^0 = 1$, and thus it follows that :

$\displaystyle \frac{a^0}{a^n} = a^{0 - n}$

Which is equivalent to :

$\displaystyle \frac{1}{a^n} = a^{-n}$

Now, you are probably going to ask us why $\displaystyle \frac{a^m}{a^n} = a^{m - n}$. To this I will answer : "Why does 1 + 0 = 1 ?".

5. Originally Posted by Bacterius
Because it is. You can explain it using this formula :

$\displaystyle \frac{a^m}{a^n} = a^{m - n}$

In our case, $\displaystyle m = 0$ because $\displaystyle a^0 = 1$, and thus it follows that :

$\displaystyle \frac{a^0}{a^n} = a^{0 - n}$

Which is equivalent to :

$\displaystyle \frac{1}{a^n} = a^{-n}$

Now, you are probably going to ask us why $\displaystyle \frac{a^m}{a^n} = a^{m - n}$. To this I will answer : "Why does 1 + 0 = 1 ?".
Thanks mate....that was a great explanation. BTW why is 1 + 0 = 1?

6. Originally Posted by rn5a
Thanks mate....that was a great explanation. BTW why is 1 + 0 = 1?
Because it is xD
It's an axiom. Mathematics are based on this assumption along with some others. You don't prove this. You assume it.

7. Originally Posted by Bacterius
Because it is xD
It's an axiom. Mathematics are based on this assumption along with some others. You don't prove this. You assume it.
Actually, the reason you don't prove it is because it's obvious. Therefore you don't assume it, you know it. Because it's obvious.

8. I wouldn't say "It's just obvious". When you have $\displaystyle 5^2$, it means $\displaystyle 1*5*5$. When the exponent, which means the number of 5s you multiply the 1 by is negative, you start dividing. Therefore, $\displaystyle 5^{-1}=\frac {1} {5}$

9. Originally Posted by Chokfull
I wouldn't say "It's just obvious". When you have $\displaystyle 5^2$, it means $\displaystyle 1*5*5$. When the exponent, which means the number of 5s you multiply the 1 by is negative, you start dividing. Therefore, $\displaystyle 5^{-1}=\frac {1} {5}$
I was talking about the 1 + 0 = 1 statement