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A cube root is a number that, when multiplied by itself three times, gives you the original number. For example, the cube root of 8 is 2 because 2 * 2 * 2 = 8. In mathematical terms, the cube root of a number 'a' is written as \(\root[3]{a}\) or \(a^{\frac{1}{3}}\).
In this exercise, you have the expression \(\root[3]{\frac{x^6}{y}}\). To simplify it, take the cube root of both the numerator (the top part of the fraction) and the denominator (the bottom part of the fraction) separately. This breaks it into two parts: \(\root[3]{x^6}\) and \(\root[3]{y}\).
Understanding how to work with cube roots helps you break down complex rational expressions into simpler parts.

Fractional exponents are another way to represent roots. Instead of writing the cube root of 'a' as \(\root[3]{a}\), you can write it as \(a^{\frac{1}{3}}\). The same rule applies to higher roots.
If you have an exponent that is a fraction, the denominator of the fraction is the root, and the numerator is the power to which the result is raised. For example, \(x^{\frac{6}{3}}\) means you take the cube root of 'x' to the sixth power, simplifying to \(x^2\).
In our exercise, to find the cube root of \( x^6\), rewrite it as \(x^{\frac{6}{3}}\), which simplifies to \(x^2\). Similarly, for \( y \), you get \(y^{\frac{1}{3}}\). Fractional exponents make it easier to work with roots within algebraic expressions.

In a fraction, the numerator is the number above the line, and the denominator is the number below the line. For instance, in the fraction \(\frac{x^6}{y}\), \(x^6\) is the numerator, and \(y\) is the denominator.
In this exercise, you are taking the cube root of both the numerator and the denominator. This means you separately apply the cube root to \(x^6\) and \(y\).
The final simplified fraction, \(\frac{x^2}{y^{1/3}}\), shows how the numerator and the denominator each transform when roots are applied to them. Working with numerators and denominators individually is a key step in simplifying rational expressions.