User:Twentysand

A common mistake when learning about exponents:

$$(a^2 + b^2)^3 \ne a^6 + b^6$$

To see why, have a look at the Binomial Theorem 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:

$${\color{Blue}(a^2 + b^2)^2} = a^4 + a^2b^2 + a^2b^2 + b^4 \, $$

So now we have

$$(a^2 + b^2)^3 = (a^4 + a^2b^2 + a^2b^2 + b^4) \times (a^2 + b^2) \, $$

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.

$$= (a^6 + 2a^4b^2 + a^2b^4 + a^4b^2 + 2a^2b^4 + b^6) \, $$