Lesson FURTHER - Radicals/Surds

Source code of 'FURTHER - Radicals/Surds'

This Lesson (FURTHER - Radicals/Surds) was created by by longjonsilver(2297)

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About longjonsilver: I have a new job in September, teaching

Introduction
After reading the "BASICS" Lesson on Radicals/Surds, there are not many more examples to consider.

However, one type of expression worth looking at is a binomial in the denominator of a fraction.

Theory
The solution of this type of question is to be aware of:

(x+a)(x-a) being the same as {{{x^2 - a^2}}} --> "difference of 2 squares".

Basically, this has 2 terms that square, which is ideal for us, since we are trying to remove (rationalise) the radical/surd.

The best explanation of these, is with an example.
Examples

Q Simplify {{{(14)/(3-sqrt(2))}}}
A We multiply the expression by "1", to keep it unchanged, but we write "1" as the second fraction in the below expression... please be happy that anything divided by itself is 1:

{{{((14)/(3-sqrt(2)))*((3+sqrt(2))/(3+sqrt(2)))}}}
{{{((14)(3+sqrt(2)))/((3-sqrt(2))(3+sqrt(2)))}}}

now, because of my choice of the second fraction, the denominator is now "correct" to be factorised using the difference of 2 squares, as:

{{{((14)(3+sqrt(2)))/((3^2-(sqrt(2))^2))}}}
{{{((14)(3+sqrt(2)))/(9-2)}}}
{{{((14)(3+sqrt(2)))/7}}}
{{{2(3+sqrt(2))}}}

Q Simplify {{{(2)/(-2+sqrt(3))}}}
A {{{(2)/(-2+sqrt(3))}}}

{{{((2)/(-2+sqrt(3)))*((-2-sqrt(3))/(-2-sqrt(3))) }}}
{{{((2(-2-sqrt(3)))/((-2+sqrt(3))(-2-sqrt(3)))) }}}
{{{(2(-2-sqrt(3)))/((-2)^2 - (sqrt(3))^2) }}}
{{{(2(-2-sqrt(3)))/(4 - 3)}}}
{{{(2(-2-sqrt(3)))/1}}}
{{{2(-2-sqrt(3))}}}

or written as {{{-2(2+sqrt(3))}}}