Question

Rationalize the denominator of each expression. Assume that all variables are positive when they appear. $$\frac{-2}{\sqrt[3]{9}}$$

Short answer

- \frac{2 \root 3 \of 81}{9}

Step-by-step solution

Identify the denominator

Observe that the denominator of the given expression is \(\frac{-2}{\root 3 \of 9}\). The goal is to rationalize this denominator.

Multiply numerator and denominator by the cube root form

Multiply the numerator and the denominator by \(\root 3 \of 81\) because \(\root 3 \of 9 \times \root 3 \of 81 = \root 3 \of {729} = 9\). The goal is to get a rational number in the denominator.

Perform the multiplication

Multiply the expression: \[ \frac{-2 \times \root 3 \of 81}{\root 3 \of 9 \times \root 3 \of 81} = \frac{-2 \root 3 \of 81}{9}\]

Simplify the expression

The expression simplifies to: \[ \frac{-2 \root 3 \of 81}{9} \]

Key concepts

rational expressions

A rational expression is a fraction with polynomials in both the numerator and the denominator. It's a way to represent a division of algebraic expressions. Just like regular fractions, we aim to simplify them when possible. This can involve factoring polynomials, cancelling out common factors, and more.

Keen students know to avoid having radicals (like cube roots) in the denominator of a rational expression. This process is called 'rationalizing the denominator'. It makes the expression simpler and easier to work with. Today, we'll explore how to rationalize with cube roots.

cube roots

Cube roots can appear tricky, yet they follow a simple principle: the cube root of a number is another number which, when raised to the power of 3, gives the original number. For example, \(\root 3 \boldsymbol {\frac{a}{b}}\) where both 'a' and 'b' can be any numbers, including variables.
Here's how to rationalize an expression involving a cube root:

• Identify the cube root in the denominator.
• Multiply both the numerator and the denominator by another factor that, when multiplied by the current denominator, provides a perfect cube (a rational number).

In our case, the denominator was \(\root 3 \boldsymbol{9}\) and we used \(\root 3 \boldsymbol{81}\) to rationalize it because \(\root 3 {9 \times 81} = \root 3 \boldsymbol{729} = 9 \). This simple multiplication allowed us to clean the expression up!

simplifying expressions

Simplifying expressions is the process of making them not only easier to understand but also easier to use in further calculations. When rationalizing the denominator, we do the following steps:

• Multiply both the numerator and the denominator by a chosen value.
• Ensure that multiplication creates a rational number in the denominator.
• Simplify the resulting expression further if possible.

In our problem, we had \(\frac{{-2}}{{\root 3 \boldsymbol{9}}}\). After multiplying by \(\root 3 \boldsymbol{81}\), this changed to \(\frac{{-2 \root 3 \boldsymbol{81}}} 9\). Breaking it down even more, it's evident that we can't simplify further because \(\root 3 81\) can't be broken down without going to decimal numbers.

Any time you work with rational expressions and cube roots, keep these steps in mind. Practice makes perfect, and soon you'll find these operations second nature!