Question 182811
<pre><font size = 4 color = "indigo"><b>

{{{(4sqrt(24))(5sqrt(18))}}}

The perferct squares are 1, 4, 9. 16. 25, 36, etc.

The largest perfect square that will divide evenly 
into 24 is 4.  So write 24 as 4*6

{{{(4sqrt(4*6))(5sqrt(18))}}}

The largest perfect square that will divide evenly 
into 18 is 9.  So write 18 as 9*2

{{{(4sqrt(4*6))(5sqrt(9*2))}}}

Take individual square roots under the radicals:

{{{( 4sqrt(4)sqrt(6) )( 5sqrt(9)sqrt(2) )}}}

We know that {{{sqrt(4)=2}}} and {{{sqrt(9)=3}}},so

{{{( 4*2sqrt(6) )( 5*3sqrt(2) )}}}

{{{( 8sqrt(6) )( 15sqrt(2) )}}}

Multiply the 8 by the 15 outside the radicals an
multiply 6 by 2 underneath the radical:

{{{( 120sqrt(12) )}}}

The largest perfect square that will divide evenly 
into 12 is 4, so write 12 as 4 times 3 

{{{( 120sqrt(4*3) )}}}

Take individual square roots under the radical:

{{{( 120sqrt(4)sqrt(3) )}}}

We know that {{{sqrt(4)=2}}}, so we have

{{{( 120*2sqrt(3) )}}}

{{{( 240sqrt(3) )}}}

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2.  {{{sqrt(5)/(sqrt(5)-2)}}}

Form the conjugate of the bottom.  That's the same
as the bottom with the sign of the second term changed.
So the conjugate of {{{sqrt(5)-2}}} is {{{sqrt(5)+2}}}
So put that over itself, as {{{(sqrt(5)+2)/(sqrt(5)+2)}}}

Now multiply the original expression by that:

{{{matrix(1,3,sqrt(5)/(sqrt(5)-2),"×", (sqrt(5)+2)/(sqrt(5)+2))}}}

Indicate the multiplication of the numerators and the denominators,
putting parentheses around those with two terms:

{{{ ( sqrt(5)(sqrt(5)+2) )/ (sqrt(5)-2)(sqrt(5)+2))   }}}

Distribute the top, and FOIL out the bottom:

{{{ ( (sqrt(5))^2+2sqrt(5) )/((sqrt(5))^2+2sqrt(5)-2sqrt(5)-4) }}}

Cancel the middle terms on the bottom:

{{{ ( (sqrt(5))^2+2sqrt(5) )/((sqrt(5))^2+cross(2sqrt(5))-cross(2sqrt(5))-4) }}}

{{{ ( (sqrt(5))^2+2sqrt(5) )/((sqrt(5))^2-4) }}}

When you square a square root you take away the radical:

{{{ ( 5+2sqrt(5) )/(5-4) }}}

{{{ ( 5+2sqrt(5) )/1 }}}

{{{  5+2sqrt(5)  }}}

Edwin</pre>