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I am currently studying about exponents and powers for college calculus discipline. In the meantime I came across negative exponents, like this $25^{-3}$ and $(5^{2})^{-3}$.

I have this calculation to solve,

$$ \frac{125^{6}\times 25^{-3}}{(5^{2})^{-3}\times25^{7}} $$

But, I get very confused and I end up getting stuck in calculations with negative exponents (i.e. $b^{-a}$) and I do not know how to solve them. So I'd like to know. How can I resolve powers with negative exponents?

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    $\begingroup$ Do you understand the definition of positive integer exponents (i.e., exponents like 2, 3, 4, etc.)? Then, do you understand the definition of negative integer exponents? What about if an exponent is zero? $\endgroup$ – MMASRP63 Aug 29 '18 at 21:54
  • $\begingroup$ @MMASRP63 positive exponent I understand and zero $0$ the number raised to 0 for results $1$, correct? $\endgroup$ – gato Aug 29 '18 at 21:59
  • $\begingroup$ Yes, you are correct. So now, do you have a definition for negative exponents as well? $\endgroup$ – MMASRP63 Aug 29 '18 at 22:00
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    $\begingroup$ For each $n \in \mathbf Z$, we have $a^{-n} = \frac{1}{a^n}$ $\endgroup$ – MMASRP63 Aug 29 '18 at 22:05
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    $\begingroup$ Have you tried searching this site for questions which are similar to yours? I want to help you with your studying, but I think you'll also benefit from consulting the resources you have (your textbook? notes? and of course this site). $\endgroup$ – MMASRP63 Aug 29 '18 at 22:06
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We have that

  • $25=5^2$
  • $125=5^3$

therefore by

  • $(a^n)^m=a^{nm}$

  • $a^n \times a^m = a^{n+m}$

  • $\frac{a^n}{a^m} = a^{n-m}$

we have

$$\frac{125^{6}\times 25^{-3}}{(5^{2})^{-3}\times25^{7}}=\frac{5^{18}\times 5^{-6}}{5^{-6}\times 5^{14}}=\frac{5^{18-6}}{5^{14-6}}=\frac{5^{12}}{5^{8}}=5^{12-8}=5^4$$

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The trick is to move negative exponents from top to bottum and vice versa.

Watch me do it: $$\frac{125^{6}\times 25^{-3}}{(5^{2})^{-3}\times25^{7}}$$

$$= \frac{125^{6}}{(5^{2})^{-3}\times25^{7}\times 25^{3}}$$

$$=\frac{(5^{2})^{3}\times125^{6}}{25^{7}\times 25^{3}}$$

$$\frac {5^6\times 5^{18}}{5^{14}\times 5^{6}}$$

$$= 5^4 = 625 $$

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