Respuesta :
Given the expressions :
[tex]\begin{gathered} (px^3+\frac{q}{x^3})^8 \\ \\ (px^2+\frac{q}{x^2})^4 \end{gathered}[/tex]The constant term in the expansions are equal
So, the constant term of the first expansion is :
[tex]^nC_r\cdot(px^3)^{n-r}\cdot(\frac{p}{x^3})^r[/tex]The constant term will appear when :
[tex]\begin{gathered} n-r=r \\ n=2r \\ r=\frac{n}{2}=\frac{8}{2}=4 \end{gathered}[/tex]So, the constant term is :
[tex]^8C_4\cdot(px^3)^4\cdot(\frac{p}{x^3})^4=70\cdot p^4\cdot q^4[/tex]And for the second expansions : n = 4
so, the constant term is :
[tex]^4C_2\cdot(px^2)^2\cdot(\frac{q}{x^2})^2=6\cdot p^2\cdot q^2[/tex]So, the constants are equal :
[tex]70\cdot p^4\cdot q^4=6\cdot p^2\cdot q^2[/tex]Divide both sides by (p^2 * q^2 )
[tex]\begin{gathered} \frac{70p^4q^4}{p^2q^2}=\frac{6p^2q^2}{p^2q^2} \\ \\ 70p^2q^2=6 \end{gathered}[/tex]Solve for p :
[tex]\begin{gathered} p^2=\frac{6}{70q^2} \\ \\ p=\sqrt[]{\frac{6}{70}}\cdot\frac{1}{q} \end{gathered}[/tex]We can write q in terms of p as following :
[tex]q=\sqrt[]{\frac{6}{70}}\cdot\frac{1}{p}[/tex]