Adding fractions with exponents

Question

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

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?

Answer 1

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: $$3^{5}\cdot 3^{-5} = 3^{0}$$.

Answer 2

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

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

Answer 3

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.