\( \sqrt { 5 } x , \sqrt[ 3 ] { 4 x ^ { 2 } y } , \sqrt[ 6 ] { 7 a ^ { 3 } b } \)

To simplify the expressions \( \sqrt{5}x \), \( \sqrt[3]{4x^2y} \), and \( \sqrt[6]{7a^3b} \), it’s helpful to rewrite them in exponential form. First, \( \sqrt{5}x \) can be expressed as \( 5^{1/2} x^{1} \). Next, \( \sqrt[3]{4x^2y} \) becomes \( 4^{1/3} x^{2/3} y^{1/3} \). Finally, \( \sqrt[6]{7a^3b} \) can be written as \( 7^{1/6} a^{3/6} b^{1/6} \) or \( 7^{1/6} a^{1/2} b^{1/6} \). Combining these different roots into one expression or evaluating them under specific scenarios could provide deeper insights or facilitate calculations. When dealing with these expressions, remember the importance of balancing the radicands and ensuring no mistakes in applying the laws of exponents. A common mistake is mixing up the indices; for example, students often miscalculate when simplifying terms that share roots. Always double-check if you're applying the correct root and whether the resulting powers match the original terms.