User:Twentysand

A common mistake when learning about exponents:
$(a^{2}+b^{2})^{3}\neq a^{6}+b^{6}$

To see why, lets expand the outer exponent:
$(a^{2}+b^{2})^{3}=(a^{2}+b^{2})\times (a^{2}+b^{2})\times (a^{2}+b^{2})$

Actually lets do this - combine the first 2 expressions back together so we can use the FOIL method. lets put it into this form and focus on the first expression(in blue):
$(a^{2}+b^{2})^{3}={\color {Blue}(a^{2}+b^{2})^{2}}\times (a^{2}+b^{2})$

We can now FOIL the first expression:
${\color {Blue}(a^{2}+b^{2})^{2}}=a^{4}+(a^{2})(b^{2})+(a^{2})(b^{2})+b^{4}\,$

So now we have
$(a^{2}+b^{2})^{3}=(a^{4}+a^{2}b^{2}+a^{2}b^{2}+b^{4})\times (a^{2}+b^{2})\,$

So we are left with a product of 2 expressions. The first expression has 4 terms, the second has 2. We use a method that is similar to FOIL, but its not exactly the same since there are 4 terms in the first expression. What we do is take the products of each combination of terms in the first expression and add them all together.
$=(a^{6}+2a^{4}b^{2}+a^{2}b^{4}+a^{4}b^{2}+2a^{2}b^{4}+b^{6})\,$