Lesson 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%5E2+-+a%5E2

--> "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 %2814%29%2F%283-sqrt%282%29%29

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:

%28%2814%29%2F%283-sqrt%282%29%29%29%2A%28%283%2Bsqrt%282%29%29%2F%283%2Bsqrt%282%29%29%29

%28%2814%29%2F%283-sqrt%282%29%29%29%2A%28%283%2Bsqrt%282%29%29%2F%283%2Bsqrt%282%29%29%29

%28%2814%29%283%2Bsqrt%282%29%29%29%2F%28%283-sqrt%282%29%29%283%2Bsqrt%282%29%29%29

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

%28%2814%29%283%2Bsqrt%282%29%29%29%2F%28%283%5E2-%28sqrt%282%29%29%5E2%29%29

%28%2814%29%283%2Bsqrt%282%29%29%29%2F%28%283%5E2-%28sqrt%282%29%29%5E2%29%29

%28%2814%29%283%2Bsqrt%282%29%29%29%2F%289-2%29

Q Simplify %282%29%2F%28-2%2Bsqrt%283%29%29

%28%282%29%2F%28-2%2Bsqrt%283%29%29%29%2A%28%28-2-sqrt%283%29%29%2F%28-2-sqrt%283%29%29%29+

%28%282%28-2-sqrt%283%29%29%29%2F%28%28-2%2Bsqrt%283%29%29%28-2-sqrt%283%29%29%29%29+

%282%28-2-sqrt%283%29%29%29%2F%28%28-2%29%5E2+-+%28sqrt%283%29%29%5E2%29+

%282%28-2-sqrt%283%29%29%29%2F%284+-+3%29

or written as -2%282%2Bsqrt%283%29%29

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