Question 83381
Hello. Can someone tell me if I am correct? in words it is 2x minus the square root of seven divided by 2x plus the square root of seven. I got -------- -1 is this correct? 2x-sq7 divided by 2x +sq7 Thanks 
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No it isn't.

{{{(2x - sqrt(7))/(2x + sqrt(7))}}}

You must multiply by {{{(2x - sqrt(7))/(2x - sqrt(7))}}}.

It's legal to multiply by that because it just equals to 1. 

{{{(2x - sqrt(7))/(2x + sqrt(7))}}}·{{{(2x - sqrt(7))/(2x - sqrt(7))}}}

or:

{{{ ((2x - sqrt(7))(2x - sqrt(7)))/((2x + sqrt(7))(2x - sqrt(7))) }}}

Now FOIL out the numerator and denominator:

{{{(4x^2 - 2x*sqrt(7) - 2x*sqrt(7) + 7)/(4x^2 + 2x*sqrt(7) - 2x*sqrt(7) - 7)}}}

The two middle terms {{{-2x*sqrt(7) - 2x*sqrt(7)}}}combine together in the 
top as {{{-4x*sqrt(7)}}}

The two middle terms {{{2x*sqrt(7) - 2x*sqrt(7)}}} CANCEL OUT in the bottom.

So we have as the final answer:

{{{(4x^2 - 4x*sqrt(7) + 7)/(4x^2 - 7)}}}

Notice that in the original expression there were radicals in BOTH
the top AND the bottom.  In the final answer, the bottom contains NO
radicals.  That's the idea -- to rewrite as an expression with NO
radicals in the bottom.  The above method causes the radicals to
cancel out in the bottom.

Edwin</pre>