Respuesta :
Given:
The sum of the cubic powers of three rational numbers is -1009.
The ratio between the first number and the second number is [tex]\dfrac{2}{3}[/tex].
The ratio between the first number and the third number is [tex]\dfrac{4}{9}[/tex].
To find:
The three numbers.
Solution:
Let [tex]x,y,z[/tex] be the three numbers, then
[tex]x^3+y^3+z^3=-1009[/tex] ...(i)
[tex]\dfrac{x}{y}=\dfrac{2}{3}[/tex]
[tex]3x=2y[/tex]
[tex]\dfrac{3}{2}x=y[/tex] ...(ii)
[tex]\dfrac{x}{z}=\dfrac{4}{9}[/tex]
[tex]9x=4z[/tex]
[tex]\dfrac{9}{4}x=z[/tex] ...(iii)
Using (i), (ii) and (iii), we get
[tex]x^3+\left(\dfrac{3}{2}x\right)^3+\left(\dfrac{9}{4}x\right)^3=-1009[/tex]
[tex]x^3+\dfrac{27}{8}x^3+\dfrac{729}{64}x^3=-1009[/tex]
[tex]\dfrac{64+216+729}{64}x^3=-1009[/tex]
[tex]\dfrac{1009}{64}x^3=-1009[/tex]
[tex]x^3=\dfrac{-1009\times 64}{1009}[/tex]
On simplification, we get
[tex]x^3=-64[/tex]
[tex]x^3=-4^3[/tex]
[tex]x=-4[/tex]
Substituting [tex]x=-4[/tex] in (ii), we get
[tex]\dfrac{3}{2}(-4)=y[/tex]
[tex]-6=y[/tex]
Substituting [tex]x=-4[/tex] in (iii), we get
[tex]\dfrac{9}{4}(-4)=z[/tex]
[tex]-9=z[/tex]
Therefore, the three numbers are -4, -6 and -9 respectively.
