(x^3 – y^3)(x^3 + y^3) =

OA : x6-y6

Take X = x^3, Y= y^3
The equation becomes (X-Y)(X+Y).
Using formula, X^2 - Y^2

Now substituting , x^6-y^6

\((a + b)(a - b) = a^2 - b^2\)
\((x^3 – y^3)(x^3 + y^3) =x^6-y^6\)

When the FOIL method is used it is clear to see that \(x^3·x^3=x^6\)
But when difference of squares is used it isn't clear whether it's x^3^2 with 3 and 2 on the same level (\(x^6\)) or x^3^2 with 2 higher than 3 (\(x^9\)). Bunuel how to ensure either case?

x^3^2 always means \(x^{3^2}\). Not sure what you mean by "3 and 2 on the same level".

Next, \(a^{m^n}=a^{(m^n)}\) and not \((a^m)^n\) (if exponentiation is indicated by stacked symbols, the rule is to work from the top down)


On the other hand \((a^m)^n=a^{mn}\).

Thank you for the quick reply. Sorry, I didn't know how to write the equations and thought the descriptions were clear enough.

\((x^3-y^3)(x^3+y^3)\)
Put in the generic difference of squares identity
\((a-b)(a+b)=a^2-b^2\)
Substitute \(x^3\) and \(y^3\) back in for a and b

How do we know that it's \(x^{3·2}\) and not \(x^{3^2}\)?

Again, I can see how it's clear using FOIL, but not how it's clear using this method. How do we know whether the exponents should be stacked?
Further, I don't think I've ever seen stacked exponents except in one GMAT Prep question (I tried to find the thread for it here to link but no luck). Not here in school learning etc. When do we see it occur in equations?

If you substitute \(a = x^3\) into \(a^2\) you get \(a^2 = (a)^2=(x^3)^2 = x^{(3*2)}=x^6\)

For stacked exponents check the following GMAT Prep question: https://gmatclub.com/forum/if-xy-1-what ... 07238.html

The given expression is in the form of (a – b) (a +b), with a = \(x^3\) and b = \(y^3\)

Using standard algebraic identities,
(a – b) (a + b) = \(a^2\) – \(b^2\)

Substituting the values of a and b in the identity, we have,

(\(x^3\) – \(y^3\)) (\(x^3\) + \(y^3\)) = \((x^3)^2\) – \((y^3)^2\)

(\(x^3\) – \(y^3\)) (\(x^3\) + \(y^3\)) = (\(x^6\)) – (\(y^6\))

The correct answer option is A.