Simplify $(576)^\frac{1}{4}(216)^\frac{1}{2}$.
To simplify this expression, I want to write both numbers as products of prime factors, using exponents to indicate repeated factors. For 576, I know that it is divisible by 2, so I can start by dividing it by 2 repeatedly until I get a power of 2. I get 576 = 2^6 * 9. For 9, I know that it is a perfect square, so I can write it as 3^2. So 576 = 2^6 * 3^2. For 216, I also know that it is divisible by 2, so I can do the same thing. I get 216 = 2^3 * 27. For 27, I know that it is a perfect cube, so I can write it as 3^3. So 216 = 2^3 * 3^3. Now I have $(576)^\frac{1}{4}(216)^\frac{1}{2} = (2^6 * 3^2)^\frac{1}{4}(2^3 * 3^3)^\frac{1}{2}$. To simplify this further, I can use the exponent rules for powers of products and powers of powers. I get $(2^6 * 3^2)^\frac{1}{4}(2^3 * 3^3)^\frac{1}{2} = 2^{\frac{6}{4}} * 3^{\frac{2}{4}} * 2^{\frac{3}{2}} * 3^{\frac{3}{2}}$. I can simplify the fractions in the exponents by dividing the numerator and denominator by their greatest common factor. I get $2^{\frac{6}{4}} * 3^{\frac{2}{4}} * 2^{\frac{3}{2}} * 3^{\frac{3}{2}} = 2^{\frac{3}{2}} * 3^{\frac{1}{2}} * 2^{\frac{3}{2}} * 3^{\frac{3}{2}}$. Now I can combine the powers of the same base by adding their exponents. I get $2^{\frac{3}{2}} * 3^{\frac{1}{2}} * 2^{\frac{3}{2}} * 3^{\frac{3}{2}} = 2^{\frac{3}{2} + \frac{3}{2}} * 3^{\frac{1}{2} + \frac{3}{2}}$. I get $2^{\frac{3}{2} + \frac{3}{2}} * 3^{\frac{1}{2} + \frac{3}{2}} = 2^3 * 3^2$. Finally, I can evaluate the powers by multiplying the base by itself the number of times indicated by the exponent. I get $2^3 * 3^2 = 8 * 9 = 72$.

