Problem 13 - Algebra A Combined Function

Question

Rationalize each denominator. Assume that all variables represent positive real numbers. \(\frac{3}{\sqrt[3]{2}}\)

Step-by-Step Solution

Answer

The rationalized expression is \( \frac{3\sqrt[3]{4}}{2} \).

Step 1: Understanding the Problem

We must rationalize the denominator in the expression \( \frac{3}{\sqrt[3]{2}} \). Currently, the denominator contains a cube root, \( \sqrt[3]{2} \), which we aim to eliminate.

Step 2: Finding Expression to Multiply

To eliminate the cube root, we multiply both the numerator and the denominator by \( \sqrt[3]{4} \), since \( \sqrt[3]{2} \times \sqrt[3]{4} = \sqrt[3]{8} = 2 \). This will rationalize the denominator.

Step 3: Multiply Numerator and Denominator

Multiply both the numerator and the denominator of the fraction by \( \sqrt[3]{4} \): \[ \frac{3}{\sqrt[3]{2}} \times \frac{\sqrt[3]{4}}{\sqrt[3]{4}} = \frac{3\sqrt[3]{4}}{\sqrt[3]{8}}.\] The denominator simplifies to 2.

Step 4: Simplify the Expression

The expression becomes \( \frac{3\sqrt[3]{4}}{2} \). This is the simplified form with the denominator rationalized.

Key Concepts

Understanding Cube Roots

Cube roots are a type of radical expression where you are looking for a number that, when multiplied by itself three times, results in the given value. In other words, the cube root of a number, say 8, is a number that, when raised to the power of 3, equals 8. This would be 2, because \(2 \times 2 \times 2 = 8\).
Just like square roots, cube roots help in expressing numbers in their simpler radical form, especially in algebraic expressions. In our task of rationalizing the denominator, we eliminate cube roots from the denominator to create a simpler, rational number.

Simplifying Expressions

Simplifying expressions refers to the process of making mathematical expressions easier to work with, mainly by reducing them into their simplest forms. This involves -

• Combining like terms
• Eliminating radicals from the denominator (rationalizing)
• Reducing fractions to their simplest form

By simplifying expressions, we make them more understandable and easier to use in further computations.
In the example of rationalizing the cube root denominator \( \frac{3}{\sqrt[3]{2}} \), simplification happens at several stages: multiplying to remove the radical and simplifying fractions to achieve the simplest form.

Understanding Algebraic Fractions

Algebraic fractions are fractions where both the numerator and the denominator contain algebraic expressions, rather than just plain numbers. These usually involve variables and require similar operations to numerical fractions, like simplifying, factoring, and rationalizing.
In the context of rationalizing denominators, when we have a denominator with a radical expression—such as a cube root—we aim to simplify or "rationalize" that denominator. This process transforms the algebraic fraction into a more straightforward form while keeping the meaning of the expression unchanged.
By doing this, we prepare the expression for easier use in further algebraic manipulations, making it more practical for students who encounter it in later calculations.