Answer:
[tex]\dfrac{x^{\frac{2}{3}}y^{\frac{1}{4}}}{4y^{2}}[/tex].
Step-by-step explanation:
The given expression is
[tex]\dfrac{3y^{\frac{1}{4}}}{4x^{-\frac{2}{3}}y^{\frac{3}{2}}\cdot 3y^{\frac{1}{2}}}[/tex]
We need to simplify the expression such that answer should contain only positive exponents with no fractional exponents in the denominator.
Using properties of exponents, we get
[tex]\dfrac{3}{4\cdot 3}\cdot \dfrac{y^{\frac{1}{4}}}{x^{-\frac{2}{3}}y^{\frac{3}{2}+\frac{1}{2}}}[/tex] [tex][\because a^ma^n=a^{m+n}][/tex]
[tex]\dfrac{1}{4}\cdot \dfrac{y^{\frac{1}{4}}}{x^{-\frac{2}{3}}y^{2}}[/tex]
[tex]\dfrac{1}{4}\cdot \dfrac{x^{\frac{2}{3}}y^{\frac{1}{4}}}{y^{2}}[/tex] [tex][\because a^{-n}=\dfrac{1}{a^n}][/tex]
[tex]\dfrac{x^{\frac{2}{3}}y^{\frac{1}{4}}}{4y^{2}}[/tex]
We can not simplify further because on further simplification we get negative exponents in numerator or fractional exponents in the denominator.
Therefore, the required expression is [tex]\dfrac{x^{\frac{2}{3}}y^{\frac{1}{4}}}{4y^{2}}[/tex].
