Question 882358
<pre>
{{{sqrt(x)=9+4sqrt(5)}}}

find the value of 

{{{sqrt(x)-1/sqrt(x)}}}

Substitute {{{9+4sqrt(5)}}} for {{{sqrt(x)}}}

{{{9+4sqrt(5)- 1/(9+4sqrt(5))}}}

Rationalize the denominator of the expression on the right
by multiplying by the conjugate of the denomintor over itself:

{{{9+4sqrt(5)- expr(1/(9+4sqrt(5)))*red(expr((9-4sqrt(5))/(9-4sqrt(5))))}}}

{{{9+4sqrt(5)- (9-4sqrt(5))/(81-16*5)}}}

{{{9+4sqrt(5)- (9-4sqrt(5))/(81-80)}}}

{{{9+4sqrt(5)- (9-4sqrt(5))/1}}}

Be careful here when you drop the 1 denominator
to enclose the numerator in parentheses because
there are two terms and the - sign preceding it
must affect both terms:

{{{9+4sqrt(5)- (9-4sqrt(5))}}}

{{{9+4sqrt(5)- 9+4sqrt(5)}}}

The 9 and -9 cancel:

{{{4sqrt(5)+4sqrt(5)}}}

{{{8sqrt(5)}}}

Edwin</pre>