Problem 15 - Intermediate Algebra

Question

Rationalize each denominator. See Examples 1 through 3. $$ \frac{3}{\sqrt[3]{2}} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Expression Type

Identify the expression type. In this exercise, we are given a fraction \( \frac{3}{\sqrt[3]{2}} \) where the denominator contains a cube root. This requires rationalization.

Step 2: Determine Rationalization Process

Since the denominator is a cube root, \( \sqrt[3]{2} \), we need to eliminate the cube root by multiplying the numerator and the denominator by the necessary power to make the denominator a perfect cube. In this case, multiply by \( \sqrt[3]{4} = (\sqrt[3]{2})^2 \) because \( (\sqrt[3]{2})^2 \times \sqrt[3]{2} = \sqrt[3]{8} = 2 \), a rational number.

Step 3: Multiply Numerator and Denominator

Multiply both the numerator and the denominator by \( \sqrt[3]{4} \). This gives:\[ \frac{3 \times \sqrt[3]{4}}{\sqrt[3]{2} \times \sqrt[3]{4}} = \frac{3 \sqrt[3]{4}}{2} \].

Step 4: Simplify the Expression

Now simplify the expression. The numerator becomes \( 3 \sqrt[3]{4} \) and the denominator is now a rational number (2), so the expression is \( \frac{3 \sqrt[3]{4}}{2} \), and it's rationalized because the denominator no longer contains a radical.

Key Concepts

Cube Roots

Cube roots answer the question: "Which number, when multiplied by itself three times, equals the given number?". For example, the cube root of 8 is 2 because multiplying 2 by itself three times results in 8: \[2 \times 2 \times 2 = 8\].
Understanding cube roots is important when dealing with numbers and expressions that involve cubed values, especially when they appear in the denominator of a fraction. Simplifying such expressions often involves rationalizing these cube roots to convert them into rational numbers. This process often includes multiplying by a form of 1 to eliminate the cube root.
This ensures that the denominator becomes a whole number, which is a preference when simplifying fractions.

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. In mathematical terms, they are often written as fractions, like \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b eq 0\).
This concept is central in rationalizing expressions because the goal is to transform irrational sections, like cube roots in denominators, into rational equivalents. This is done to create a standardized form that's easier to interpret and use.
Converting an expression like \(\frac{3}{\sqrt[3]{2}}\) into a rational number involves transforming it into an equivalent expression with a rational denominator, like \(\frac{3\sqrt[3]{4}}{2}\).

Fraction Simplification

Simplifying fractions involves reducing them to their simplest form so that both the numerator and the denominator are as small as possible while retaining their original ratio. This can involve:

• Reducing common factors from both the numerator and the denominator.
• Making the denominator a whole number by rationalizing it, as seen with radicals or roots.

Once a fraction is simplified, it is often much easier to work with in mathematical problems or equations. A critical part of simplifying is ensuring that the fraction represents the same quantity as the original but in a more manageable form.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators (like addition and multiplication). They form the basis of algebra and can represent a wide range of mathematical problems.
When dealing with rationalization, you'll often encounter algebraic expressions. Rationalizing requires using algebraic knowledge to manipulate and simplify these expressions, especially when they involve roots or powers.
For example, in rationalizing \(\frac{3}{\sqrt[3]{2}}\), algebraic manipulation plays a vital role in transforming the expression into \(\frac{3\sqrt[3]{4}}{2}\). Here, the understanding of multiplying powers and roots helps in appropriately eliminating radicals from the denominator.