Answer:
x = y = - 2
Step-by-step explanation:
Using the rule of radicals/ exponents
[tex]a^{\frac{m}{n} }[/tex] = [tex]\sqrt[n]{a^{m} }[/tex] , [tex]a^{-m}[/tex] = [tex]\frac{1}{a^{m} }[/tex] , [tex](a^m)^{n}[/tex] = [tex]a^{mn}[/tex]
Given
[tex]\sqrt[x]{4}[/tex] = [tex]\frac{1}{2}[/tex] , then
[tex]\sqrt[x]{2^{2} }[/tex] = [tex]\frac{1}{2}[/tex]
[tex]2^{\frac{2}{x} }[/tex] = [tex]2^{-1}[/tex]
Since the bases on both sides are equal, both 2 , then equate exponents
[tex]\frac{2}{x}[/tex] = - 1 ( multiply both sides by x )
2 = - x , that is
x = - 2
Similarly
[tex]\sqrt[y]{9}[/tex] = [tex]\frac{1}{3}[/tex] , then
[tex]\sqrt[y]{3^{2} }[/tex] = [tex]\frac{1}{3}[/tex]
[tex]3^{\frac{2}{y} }[/tex] = [tex]3^{-1}[/tex]
Equating the exponents gives
[tex]\frac{2}{y}[/tex] = - 1 ⇒ - y = 2 ⇒ y = - 2
