Question

Part 1: Explain, in complete sentences, the effect of the difference of squares pattern on the multiplication of radicals. (2 points)

Part 2: Give an example. (2 points)

Answer 1

1.) Part 1:
The product of a sum and a difference of two values is equal to the difference of their squares.
If both of those quantities are square roots of integers, for example p and q
such that a = √p and b = √q
then performing the expansion squares the radicals leaving only integers.
(√p + √q)(√p - √q) = (a + b)(a - b) = a^2 - b^2 = p - q

If only one of the quantities is a square root of an integer we get
(p + √q)(p - √q) = (a^2 + b)(a^2 - b) = a^4 - b^2 = p^2 - q

Part 2: For these examples let a = 3, b = 7
(√3 + √7)(√3 - √7) = (√3)^2 - (√7)^2 = 3 - 7 = - 4, and
(3 + √7)(3 - √7) = (3)^2 - (√7)^2 = 9 - 7 = +2