Solution
To find the values of H and K, we need to write
[tex]x^3y^9\sqrt{xy}[/tex]
into
[tex]\sqrt{x^Hy^K}[/tex]
Note that
[tex]\begin{gathered} x^3=\sqrt{(x^3})^2=\sqrt{x^6} \\ \\ y^9=\sqrt{(y^9)^2}=\sqrt{y^{18}} \end{gathered}[/tex]
From the question, we have
[tex]\begin{gathered} x^3y^9\sqrt{xy}=x^3\times y^9\times\sqrt{xy} \\ x^3y^9\sqrt{xy}=\sqrt{x^6}\times\sqrt{y^{18}}\times\sqrt{xy} \\ x^3y^9\sqrt{xy}=\sqrt{x^6\times y^{18}\times x\times y} \\ x^3y^9\sqrt{xy}=\sqrt{x^7y^{19}} \end{gathered}[/tex]
Therefore,
[tex]\begin{gathered} H=7 \\ K=19 \end{gathered}[/tex]
