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$$3^5 + {1\over3^5}=?$$

My first instinct was to rewrite the second term as $3^{-5}$. Since the base is $3$, rewrite as $3^{5+-5}$. It simplifies to $3^0= 1$. Apparently this is incorrect. Can anyone please explain why?

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This is incorrect because the exponent rule that you were thinking of is: $$a^b\cdot a^c = a^{b + c}$$ So if you had $3^{5}\cdot 3^{-5}$ then you could use that rule: $$3^{5}\cdot 3^{-5} = 3^{0}$$.

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  • $\begingroup$ Oh okay, thanks. So I can only add the exponents when the bases are being multiplied? Is there a way to simplify bases when it involves addition? $\endgroup$ – Emma Apr 13 '15 at 15:15
  • $\begingroup$ @Emma The short answer is no. $\endgroup$ – DRF Apr 13 '15 at 15:16
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As has already been pointed out $a^b+a^c\neq a^{b+c}$. That means in your case the best you can do is try and get a common denominator for both numbers and add them. Try and see if you know how.

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There is not much you can do in terms of simplifying these exponents. You could rewrite $3^{5}$ as $\frac{3^{10}}{3^{5}}$ then simplify with common denominators to

$$\frac{3^{10}+1}{3^{5}}$$

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