Beginner exponent/simplification question

Question

Hey there I am having some trouble remembering all the old exponent rules and such, for example, $$ \frac{1}{(6+7^n) ^3} $$

How can I simplify this? I know that (7^n)^3 is the same as (7^3n), but what about the 6 inside the brackets?

Thanks all

Edit: The reason I am asking in this question is in regard to a problem of the following type: Given a sequence An, find some number R such that An/R^n has a finite non zero limit. The reason this is useful is because if this is true than using the limit comparison test, we are able to conclude if either both converge, or both diverge.

Answer 1

I think $(6+7^n)^3$ is as simple as it is going to get. Unfortunately, it is seldom true that $(a+b)^k = a^k+b^k$ for any numbers $a,b$ with exponent $k$. If you have to change the denominator, you could calculate by hand $$(6+7^n)(6+7^n)(6+7^n)$$ or by using the binomial theorem. But just leaving it as $(6+7^n)^3$ seems like the best option.

Answer 2

The only thing you can do is expanding, rather than simplifying:

$$(6+7^n)^3 = 6^3+3\cdot 6^2 \cdot 7^n+3\cdot 6\cdot 7^{2n}+1\cdot 7^{3n}$$

Answer 3

You should use newtons binom and think of $7^n$ as just $b$ from $(a+b)^3 = (a^3 + 3a^2b+3ab^2+b^3)$