following without the root sign. \( \sqrt[4]{a^{6}} \) (b) \( \sqrt[3]{x^{5}} \) (c) \( \sqrt{a^{5}} \) following with a root sign.

For the expression \( \sqrt[4]{a^{6}} \), you can rewrite it as \( a^{6/4} \) or simplified to \( a^{3/2} \). This is a neat way to handle roots, essentially telling you how many times you'd multiply \( a \) together to achieve that power! In the case of \( \sqrt[3]{x^{5}} \), this transforms to \( x^{5/3} \). This shows you that if you were to take the cube root of \( x^5 \), you're finding a value that raises \( x \) to the power of \( 5/3 \), which is like multiplying \( x \) together slightly more than once. For \( \sqrt{a^{5}} \), you convert it into \( a^{5/2} \), which is equivalent to \( a^2 \) times \( \sqrt{a} \). This expression shows how to mix whole numbers and roots, allowing you to see not only the integer part but also the remaining fraction.