If $(\sqrt{2})^x+(\sqrt{3})^x=(\sqrt{13})^{x/2}$, then the number of real values of x is _____.
Answer & explanation
We have,
$(\sqrt{2})^x+(\sqrt{3})^x=(\sqrt{13})^{x/2}$
$⇒2^{x/2}+3^{x/2}=(\sqrt{13})^{x/2}$
$⇒(\frac{2}{\sqrt{13}})^{x/2}+(\frac{3}{\sqrt{13}})^{x/2}=(\frac{2}{\sqrt{13}})^{2}+(\frac{3}{\sqrt{13}})^2$
$⇒\frac{x}{2}=2$ [On comparison]
$⇒x=4$