Question 1201476
<br>
{{{x^24=w}}} --> {{{x=w^(1/24)}}}<br>
{{{y^40=w}}} --> {{{y=w^(1/40)}}}<br>
Then<br>
{{{xyz^12=w}}}
{{{(w^(1/24))(w^(1/40))(z^12)=w}}}
{{{(w^(5/120))(w^(3/120))(z^12)=w}}}
{{{(w^(8/120))(z^12)=w}}}
{{{(w^(1/15))(z^12)=w}}}
{{{z^12=w^(14/15)}}}
{{{z=w^(14/180)=w^(7/90)}}}<br>
The system of equations is indeterminate; there is an infinite family of solutions.<br>
ANSWERS:<br>
{{{w=p^90)}}} where p is any number (except 1, according to the problem description)<br>
{{{x=w^(1/24)=p^(90/24)=p^(15/4)}}}
{{{y=w^(1/40)=p^(90/40)=p^(9/4)}}}
{{{z=w^(7/90)}}}
{{{z^12=w^(84/90)=p^84}}}<br>
CHECK:
{{{xyz^12=((p^(15/4))(p^(9/4))(p^84))=p^90=w}}}<br>

x^24=w --> x=w^(1/24)<br>
y^40=w --> y=w^(1/40)<br>
Then<br>
xyz^12=w
(w^(1/24))*(w^(1/40))*(z^12)=w
(w^(5/120))*(w^(3/120))*(z^12)=w
(w^(8/120))*(z^12)=w
(w^(1/15))*(z^12)=w
z^12=w^(14/15)
z=w^(14/180)=w^(7/90)<br>
The system of equations is indeterminate; there is an infinite family of solutions.<br>
ANSWERS:<br>
w=p^90 where p is any number (except 1, according to the problem description)<br>
x=w^(1/24)=p^(90/24)=p^(15/4)
y=w^(1/40)=p^(90/40)=p^(9/4)
z=w^(7/90)
z^12=w^(84/90)=p^84<br>
CHECK:
xyz^12=((p^(15/4))*(p^(9/4))*(p^84))=(p^6)(p^84)=p^90=w<br>