So I think I need some clarification about the rules for manipulating indices, in particular these two equivalences:
$(x^3)^2 = x^{(3)(2)} = x^6$
$a = a^1$
Take the expression: $(5+5)^2$, which is equivalent to $(5+5)(5+5) = 100$.
Using the rules for indices above, I would have thought that I could use them to do this: $(5^1+5^1)^2 = 5^{(1)(2)}+5^{(1)(2)} = 5^2+5^2$, but this is obviously wrong as $5^2+5^2 = 50 \neq 100$.
Where am I going wrong with my application of the rules for indices?
Your error is here: $$ (5^1+5^1)^2 \color{red}{\neq} 5^{(1)(2)}+5^{(1)(2)}. $$ You can easily verify that these two numbers are not the same. In general, we have $(a+b)^2=a^2+b^2+2ab$, so it is typically the case that $(a+b)^2\neq a^2+b^2$. This is true: $$ (5^1+5^1)^2 \color{red}{=} 5^{(1)(2)}+5^{(1)(2)}+2\cdot5^{1}\cdot5^{1}. $$
Addendum: Your "rule" $(a+b)^n=a^n+b^n$ does not follow from the two rules you stated in your post. The two rules you mentioned say nothing about the power of a sum. To calculate the power of a sum, you need something more. In the case $n=2$ we get, using the definition $x^2=x\cdot x$, that $$ (a+b)^2=(a+b)(a+b)=a^2+ab+ba+b^2=a^2+2ab+b^2. $$ You can get similar formulas for other $n$s in a similar way; see the binomial theorem.