# Adding fractions with exponents

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

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?

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: $$3^{5}\cdot 3^{-5} = 3^{0}$$.
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.
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
$$\frac{3^{10}+1}{3^{5}}$$