why is there no absolute value symbols in the difference of perfect squares formula?

Question

We already know that $ \sqrt{b^2} = | b| $. For the difference of perfect squares formula $ a^2 - b^2 = (a+b)(a-b) $, why is there no absolute value symbols in this formula? like this $ a^2 - b^2 = ( |a| + |b| )( |a| - |b|) $

Answer 1

The absolute values aren't there because they aren't needed. There are squares but no square roots in the problem. The identity is just the distributive law three times and the commutative law:

$$ (a+b)(a-b) = a(a-b) + b(a-b) = a^2 - ab + ba - b^2 = a^2 - b^2. $$

The absolute values are correct when $a$ and $b$ are real numbers, but the identity is true in much more generality, where absolute values might not make sense. You should check it for polynomials: suppose $a$ is the expression $x+2$ and $b$ is $x^2-3x+1$.

Answer 2

(Promoted from a comment.)

We do know that $\sqrt{b^2}=|b|$ for real numbers (but not for complex numbers). And from that we also get that $b^2=|b|^2$. Therefore, for real numbers, we can write: $$(a+b)(a-b)=a^2-b^2=|a|^2-|b|^2=(|a|+|b|)(|a|-|b|)$$ and therefore for real numbers your identity $$a^2-b^2=(|a|+|b|)(|a|-|b|)$$ is entirely correct!

Of course the identity $a^2-b^2=(a+b)(a-b)$ is valid in every commutative ring (even where no kind of absolute value is defined).