\( \left. \begin{array} { l } { \sqrt[ 4 ] { 16 ^ { 3 } } } \\ { \sqrt[ 3 ] { 64 ^ { 2 } } } \\ { ( \frac { 81 } { 16 } ) ^ { \frac { 1 } { 4 } } } \\ { ( 0.0625 ) ^ { \frac { 1 } { 4 } } } \\ { ( 0,25 x ^ { 4 } y ^ { 8 } ) ^ { \frac { 1 } { 4 } } } \end{array} \right. \)

To break it down in a fun way, let’s unravel these mathematical expressions! First up, \( \sqrt[4]{16^3} \) is essentially asking for the fourth root of \( 16^3 \). Since \( 16 \) is \( 2^4 \), we can rewrite \( 16^3 \) as \( (2^4)^3 = 2^{12} \). Taking the fourth root gives us \( 2^{12/4} = 2^3 = 8 \). Now, moving on to \( \sqrt[3]{64^2} \), we know \( 64 \) is \( 4^3 \). Thus, \( 64^2 \) can be expressed as \( (4^3)^2 = 4^6 \). Remembering that 64 is actually \( 2^6 \) makes it \( (2^6)^2 = 2^{12} \). The cube root then turns into \( 2^{12/3} = 2^4 = 16 \). Next, \( \left(\frac{81}{16}\right)^{\frac{1}{4}} \) is a charm. With \( 81 = 3^4 \) and \( 16 = 2^4 \), this simplifies down to \( (3^4 / 2^4)^{\frac{1}{4}} = \frac{3^{4/4}}{2^{4/4}} = \frac{3}{2} \). For \( (0.0625)^{\frac{1}{4}} \), knowing that \( 0.0625 = \frac{1}{16} = \left(\frac{1}{2}\right)^4 \), we see that taking the fourth root gives us \( (0.0625)^{\frac{1}{4}} = 0.5 \). Finally, \( (0.25x^4y^8)^{\frac{1}{4}} \) is the last equation. With \( 0.25 = \frac{1}{4} = \left(\frac{1}{2}\right)^2 \), we can express the entire equation as: \( (0.25)^{1/4}(x^4)^{1/4}(y^8)^{1/4} = (2^{-2})^{1/4}(x)^{1}(y^2) = \frac{1}{\sqrt{2}}xy^2 \). Math is like a treasure hunt, where every step reveals a shiny nugget of knowledge!