<s>[INST] Law of Indices and Quadratic Expressions 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?
 [/INST] 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.
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