Answer:
because x,y,z are in continuous proportion
=> x/y = y/z
<=> xz = y² =>
[tex]\frac{xz}{y} =y\\\\=>\frac{x^{3}z^{3} }{y^{3} }=y^{3}[/tex](1)
with (1), we have:
[tex]x^{2}y^{2}z^{2}(\frac{1}{x^{3} }+\frac{1}{y^{3} }+\frac{1}{z^{3} })\\\\= (xyz)^{2}(\frac{1}{x^{3} } +\frac{y^{3} }{x^{3}z^{3} }+\frac{1}{z^{3} })\\\\=(xyz)^{2} (\frac{x^{3}+y^{3}+z^{3} }{x^{3}z^{3}} )\\\\=y^{2}.\frac{x^{3}+y^{3}+z^{3} }{xz} \\\\= xz.\frac{x^{3}+y^{3}+z^{3} }{xz} \\\\=x^{3}+y^{3}+z^{3}[/tex]
Step-by-step explanation:
