Explain, in complete sentences, the effect of the difference of squares pattern on the multiplication of radicals. Give an example.

Question

terenzreignz · Answer

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
$$a ^{2} - (\sqrt{b})^{2} = a^{2} - b$$

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

$$(2 + \sqrt{3})(2 - \sqrt{3}) = 2^{2} - (\sqrt{3})^{2} = 4 - 3 = 1$$