ordering radical and exponential expressions

1. Convert all expressions to a common exponent base: To compare the values, we first express them all using fractional exponents. The given values are $\sqrt[3]{9}$, $4^{\frac{2}{3}}$, and $3^{\frac{1}{2}}$.

2. Rewrite the expressions:
- $\sqrt[3]{9} = 9^{\frac{1}{3}} = (3^2)^{\frac{1}{3}} = 3^{\frac{2}{3}}$
- $4^{\frac{2}{3}} = (2^2)^{\frac{2}{3}} = 2^{\frac{4}{3}}$
- $3^{\frac{1}{2}}$

3. Find a common denominator for the exponents: The exponents are $\frac{2}{3}$, $\frac{4}{3}$, and $\frac{1}{2}$. The least common denominator is 6. We rewrite the exponents:
- $3^{\frac{2}{3}} = 3^{\frac{4}{6}} = (3^4)^{\frac{1}{6}} = 81^{\frac{1}{6}}$
- $2^{\frac{4}{3}} = 2^{\frac{8}{6}} = (2^8)^{\frac{1}{6}} = 256^{\frac{1}{6}}$
- $3^{\frac{1}{2}} = 3^{\frac{3}{6}} = (3^3)^{\frac{1}{6}} = 27^{\frac{1}{6}}$

4. Compare the bases: Since all expressions are now in the form $x^{\frac{1}{6}}$, we simply compare the bases: $27$, $81$, and $256$.
- $27 < 81 < 256$
- Therefore, $27^{\frac{1}{6}} < 81^{\frac{1}{6}} < 256^{\frac{1}{6}}$

1. Final ordering: Substituting back the original values, we get $3^{\frac{1}{2}} < \sqrt[3]{9} < 4^{\frac{2}{3}}$.