Which is equivalent to RootIndex 4 StartRoot 9 EndRoot Superscript one-half x?

The expression for which you want to find an equivalent form is:\({(\sqrt[4]{9})}^{\frac{1}{2}x}\)Answer:Here there are 3 equivalent expressions:\({(\sqrt[4]{3^2})}^{\frac{1}{2}x}\)\((\sqrt{3})^{\frac{1}{2}x}\)\((\sqrt[4]{3} )^x\)Explanation:There are many equivalent forms that you can find, so I will show some of the most important.The first step that you should do is to put the radicand number (the number inside the root, i.e. 3) as a product of its prime factors. Thus the first equivalent expression is:\({(\sqrt[4]{3^2})}^{\frac{1}{2}x}\)Now you can simplify the root index, 4, with the exponent of the radicand, 2, an you get a new equivalent:\({(\sqrt[4]{3^2})}^{\frac{1}{2}x}=(\sqrt{3})^{\frac{1}{2}x}\)Another equivalent form is obtained if you convert the 1/2 index (before the x) into a root index:\((\sqrt{3})^{\frac{1}{2}x}=(\sqrt{\sqrt{3}})^x=(\sqrt[4]{3} )^x\)The main properties used to find those equivalent expressions are:\(a^{\frac{1}{n}}=\sqrt[n]{a}\)And:\({(a^{\frac{1}{n}})^m=a^{\frac{m}{n}}= \sqrt[n]{a^m}\)