Answer
$\displaystyle \frac{x^{2}}{7y^{3/2}}$
Work Step by Step
Note: assuming all variables are positive, we do not need the absolute value brackets.
For positive $a,$
$(a^{2})^{1/2}=\sqrt{a^{2}}=|a|$, the square root is $a.$
$(49x^{-2}y^{4})^{-1/2}(xy^{1/2})=\qquad$... apply $(ab)^{n}=a^{n}b^{n}$
$=(49)^{-1/2}(x^{-2})^{-1/2}(y^{4})^{-1/2}(xy^{1/2})\qquad$... apply $(a^{m})^{n}=a^{mn}$
$=(49)^{-1/2}(x^{-2\cdot(-1/2)})y^{4(-1/2)}(xy^{1/2})\qquad$... apply $a^{-n}=\displaystyle \frac{1}{a^{n}}$
$=\displaystyle \frac{1}{49^{1/2}}x^{1}y^{-2}\cdot x^{1}y^{1/2}\qquad$... apply $a^{m}a^{n}=a^{m+n}$
$=\displaystyle \frac{1}{7}x^{1+1}y^{-2+(1/2)}$
$=\displaystyle \frac{1}{7}x^{2}y^{-3/2}\qquad$... apply $a^{-n}=\displaystyle \frac{1}{a^{n}}$
$=\displaystyle \frac{x^{2}}{7y^{3/2}}$
