91Ó°ÊÓ

Radicals are symbols that represent the root of a number. The most common radicals are square roots (\( \sqrt{ } \)) and cube roots (\( \sqrt[3]{ } \)). In this exercise, we deal with a cube root. The cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, \( \sqrt[3]{8} = 2\), because \( 2 \times 2 \times 2 = 8 \). To rationalize a denominator with a cube root, you want to eliminate the radical symbol. This often involves multiplying by an expression that, when added to the current denominator, forms a power that's a multiple of 3. This transforms the radical into a simpler expression, making calculations easier and the expression cleaner.

The denominator is the bottom part of a fraction. When rationalizing, our goal is to remove radicals from the denominator. In the given exercise, the denominator starts as \(ab^2\). Rationalizing a denominator with a cube root involves creating an expression where the denominator is a whole number or a simpler term. We achieve this by multiplying both the numerator and the denominator by a term that will make the denominator a perfect power of 3, which eliminates the radical. Through this process, the denominator becomes more manageable and the entire expression is simplified.

A cube root of a number \(x\) is a number \(y\) such that \(y^3 = x\). For example, cube root of 27 is 3 because \(3^3 = 27\). In the exercise, the cube root in the expression affects both the numerator and the denominator. To rationalize the denominator, one must transform it into a perfect cube. This often involves multiplying by an additional factor that complements the existing term. Here, we used \( \sqrt[3]{a^2 b} \) to ensure that the denominator becomes \(a^3 b^3\). This simplifies to \(ab\), removing the cube root and making the fraction easier to handle.