# What does “$\to$” mean in “$a^3+b^3=(a+b)^3\rightarrow ab(a+b)=0$”?

I'm working on this problem and I'm having difficulty understanding the solution. In the solution, it states $$a^3+b^3=(a+b)^3\rightarrow ab(a+b)=0$$. What do these equations mean? It doesn't seem to fit the identity $$a^3+b^3=(a+b)(a^2-ab+b^2)$$ if the arrow meant subtracting.

I know that right arrows can define a function, but I don't see how $$(a+b)^3\rightarrow ab(a+b)=0$$ could be a function. If it is, then I don't know how it is applicable to the problem, in which case it would be great if someone could explain the rest of the solution.

P. S. I don't see this as asking two questions if that's not allowed. One is just a prerequisite to the other.

• The argument is asserting that $a^3+ b^3 = ( a+b)^3$, then indicating that this relation leads to ("$\to$") $ab(a+b)=0$. That kind of notation probably makes more sense in displayed equations like $$a^3+ b^3 = ( a+b)^3 \qquad\to\qquad ab(a+b)=0$$ (At least, I hope it makes sense ... I do this all the time!) Incorporating it into text probably isn't the best presentation. – Blue Nov 9 '19 at 5:20
• $“A\to B”$ means that whenever $A$ is true, then $B$ is also true. – MJD Nov 9 '19 at 15:11

Assume that $$a^3 + b^3 = \left( {a + b} \right)^3$$ Since $$\left( {a + b} \right)^3 = a^3 + b^3 + 3a^2 b + 3ab^2$$ it follows that $$3a^2 b + 3ab^2 = 0$$ thus $$3ab\left( {a + b} \right) = 0$$ so that $$ab\left( {a + b} \right) = 0$$

It means: if $$a^3+b^3 =(a+b)^3$$, then $$ab (a+b)=0$$.

In this context, the arrow is a symbol which describes the logical flow of the argument being made. It should be read as "implies" or "leads to". That is, if $$A$$ and $$B$$ are two statements, then the notation $$A \to B$$ can be read "$$A$$ implies $$B$$", or "$$A$$ leads to $$B$$". Alternatively, this can also be read as "If $$A$$ is a true statement, then $$B$$ is a true statement as well", or, more succinctly, "If $$A$$, then $$B$$".

In the context of the particular argument presented in the question, the notation $$a^3+b^3=(a+b)^3\rightarrow ab(a+b)=0$$ means that the truth of the first equation implies the truth of the second equation. Written with sightly less notation, we might say

If $$a^3 + b^3 = (a+b)^3$$, then $$ab(a+b)=0$$.

This answer shows how one might obtain that result. Written more compactly, one might write

\begin{align} &a^3 + b^3 = (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 && \text{(expand the RHS)} \\ &\qquad\rightarrow 0 = 3a^2b + 3ab^2 = 3ab(a^2+b^2) && \text{(cancel a^3+b^3, factor)} \\ &\qquad\rightarrow 0 = ab(a^2 + b^2). && \text{(cancel 3)} \end{align}

the arrow means it follows: just calculate (a+b)^3 and subtract a^3+b^3 what remains is 0

• This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review – Klangen Nov 9 '19 at 5:45
• @Klangen, I think it does answer the question. The arrow means from the first it follows the second. One must expand the LHS and substract the RHS from it. – farruhota Nov 9 '19 at 7:51