Question

How do you multiply `[a^(1/2) + 6^(1/2)] [a^(1/2) - 6^(1/2)]`?

Answer 1

You could multiply out out using FOIL, or whatever idea you've been taught for multiplying binomials.

It's quicker to recognize that this product is of the form:

`(x+y)(x-y) = x^2-y^2)`

(The product of a sum and difference equals the difference of the squares.)

Now realize that `(a^(1/2))^2 = a` and `(6^(1/2))^2 = 6`, so the product is:

`[a^(1/2) + 6^(1/2)] [a^(1/2) - 6^(1/2)] =(a^(1/2))^2 - (6^(1/2))^2 = a-6`

OR

`[a^(1/2) + 6^(1/2)] [a^(1/2) - 6^(1/2)] =a^(1/2)a^(1/2) - a^(1/2)6^(1/2)+a^(1/2)6^(1/2)-6^(1/2)6^(1/2) = a^(1/2+1/2) - 6^(1/2+1/2) = a-6`