Question

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

Part 2: Give an example.

Answer 1

If, for instance, we have
\[a+\sqrt{b}\]
and we multiply it with its conjugate
\[a-\sqrt{b}\]
where a is a real number and b is a positive number which is not a perfect square, then
all we need remember is the pattern of "difference of two squares", like so :
\[(x+y)(x-y)=x^2-y^2\]
In our case,
x=a
and
\[y=\sqrt{b}\]
So the result of multiplying them is
\[a^2-(\sqrt{b})=a^2-b\]
In other words, the effect of the "difference of two squares" pattern is the loss of the radical sign, typically used to simplify equations.
For example
\[(2+\sqrt{3})(2-\sqrt{3})=2^2-(\sqrt{3})^2=4-3=1\]