Question 165128
Use six unit multipliers to convert 6 cubic yards to cubic inches
<pre><font size = 4 color = "indigo"><b>
     Start by writing {{{6_cubic_yards}}} as {{{6yd^3}}} and
     then as {{{6*yd*yd*yd}}}, and then put it over {{{1}}}, 
     and we start with this:

{{{(6*yd*yd*yd)/1}}}

     We treat a unit just as we would treat a number or
     letter in algebra which is multiplied by a number. So we 
     want to cancel away the three {{{yd}}} factors and bring 
     in {{{ft}}}.  We know that {{{3ft=1yd}}}.  So there are
     two possible unit multipliers which we can make with
     that.  We can make the unit multiplier {{{(3ft)/(1yd)}}}
     or the unit multiplier {{{(1yd)/(3ft)}}}.  The question
     is which one do we make.  Since we have {{{yd}}} in the
     numerator, we must use the unit multiplier which has
     {{{yd}}} in the denominator so the {{{yd}}}'s will cancel.
     So we multiply by {{{(3ft)/(1yd)}}} three times, one for
     each {{{yd}}} factor.  

{{{matrix(1,7,(6*yd*yd*yd)/1, "×",(3ft)/(1yd),"×",(3ft)/(1yd),"×",(3ft)/(1yd)) }}} 

     and then the {{{yd}}}'s cancel, like this:

{{{matrix(1,7,(6*cross(yd)*cross(yd)*cross(yd))/1, "×",(3ft)/(1cross(yd)),"×",(3ft)/(1cross(yd)),"×",(3ft)/(1cross(yd)) ) }}} 



     Now we want to cancel away the three {{{ft}}} factors
     and bring in {{{in}}}.  We know that {{{12in=1ft}}}. 
     So there are two possible unit multipliers which we can 
     make with that.  We can make the unit multiplier {{{(1ft)/(12in)}}}
     or the unit multiplier {{{(12in)/(1ft)}}}.  The question
     is which one do we make.  Since we have {{{ft}}} in the
     numerator, we must use the unit multiplier which has
     {{{ft}}} in the denominator so the {{{ft}}}'s will cancel.
     So we multiply by {{{(12in)/(1ft)}}} three times, one for
     each {{{ft}}} factor.  

{{{matrix(1,13,(6*cross(yd)*cross(yd)*cross(yd))/1, "×",(3ft)/(1cross(yd)),"×",(3ft)/(1cross(yd)),"×",(3ft)/(1cross(yd)), "×", (12in)/(1ft), "×", (12in)/(1ft), "×", (12in)/(1ft) ) }}} 

     Then the {{{ft}}}'s cancel:

{{{matrix(1,13,(6*cross(yd)*cross(yd)*cross(yd))/1, "×",(3cross(ft))/(1cross(yd)),"×",(3cross(ft))/(1cross(yd)),"×",(3cross(ft))/(1cross(yd)), "×", (12in)/(1cross(ft)), "×", (12in)/(1cross(ft)), "×", (12in)/(1cross(ft)) ) }}}

     So all we have left is:

{{{6*3*3*3*12*12*12*in*in*in }}}

     Multiply all that out with a calculator,
     and replace the {{{in*in*in}}} by {{{in^3}}}

{{{279936*in^3}}}

     or 279936 cubic inches.

     Edwin</pre>