Problem 82 - A Graphical Approach to College Algebra

Question

Rationalize the denominator of each radical expression. Assume that all variables represent nonnegative real numbers and that no denominators are \(0 .\) $$\frac{4}{\sqrt[3]{a^{2}}}$$

Step-by-Step Solution

Answer

The rationalized expression is \( 4a^{-2/3} \).

Step 1: Understanding the Problem

The problem is to rationalize the denominator of the fraction \( \frac{4}{\sqrt[3]{a^{2}}} \). Rationalizing the denominator involves rewriting the expression so that the denominator is a whole number (without a radical).

Step 2: Expressing the Radical as Fractional Exponent

Rewrite the denominator's radical using fractional exponents: \( \sqrt[3]{a^{2}} = a^{2/3} \). The expression becomes \( \frac{4}{a^{2/3}} \).

Step 3: Identify the Rationalizing Factor

To eliminate the cube root in the denominator, raise \( a^{2/3} \) to a power that results in a whole number exponent. We need to multiply by \( a^{1/3} \) so that the exponent becomes 1: \( a^{2/3} \times a^{1/3} = a^{3/3} = a^{1} \).

Step 4: Multiply by the Rationalizing Factor

Multiply both the numerator and the denominator by \( a^{1/3} \):\[\frac{4}{a^{2/3}} \times \frac{a^{1/3}}{a^{1/3}} = \frac{4 \cdot a^{1/3}}{a^{2/3} \cdot a^{1/3}}\].

Step 5: Simplify the Expression

Simplify the denominator: \( a^{2/3} \times a^{1/3} = a^{3/3} = a^{1} \). The expression becomes \( \frac{4a^{1/3}}{a} \).

Step 6: Final Formulation

Rewrite the result as \( 4 \times a^{1/3 - 1} = 4a^{-2/3} \). Thus, the final expression with the denominator rationalized is \( \frac{4}{a^{2/3}} = 4a^{-2/3} \).

Key Concepts

Radical Expressions

In mathematics, radical expressions involve roots, such as square roots, cube roots, and higher-order roots. These expressions include a radical symbol, which indicates the root being taken. A common goal when working with radicals is to simplify the expression so that it's easier to understand or compute. When simplifying radical expressions, we often focus on the denominator to remove the radical. This process is known as "rationalizing the denominator."Here's why this is important:

• Standardization: Mathematicians prefer to have denominators without radicals for consistency and ease of comparison.
• Simplification: It makes further operations like addition, subtraction, or multiplication simpler.
• Lowers complexity in solving equations that involve fractions with radicals.

In our problem, the denominator involves a cube root, expressed as \(\sqrt[3]{a^2}\). To rationalize it, we need to convert it into a form that does not involve the radical. Understanding this is a foundational step in mastering algebraic manipulation.

Fractional Exponents

Fractional exponents offer a powerful way to express radicals in algebra. Instead of using a radical symbol, we employ fractional notation to denote roots. This makes performing operations easier, as it transforms the operation into multiplication.To understand fractional exponents:

• The denominator of the fractional exponent indicates the root. For example, \(a^{1/3}\) represents the cube root of \(a\).
• The numerator specifies the power to which the base is raised. Thus, \(a^{2/3}\) means the cube root of \(a^2\).

Using fractional exponents, we convert \(\sqrt[3]{a^2}\) to \(a^{2/3}\). This transformation allows us to manipulate the expression more freely using the laws of exponents. By understanding this conversion, we can strategically simplify and rationalize expressions with radicals in a structured manner.

Algebraic Simplification

Algebraic simplification is about reducing expressions to their simplest form. It's an essential skill in algebra that helps in solving equations, comparing functions, and understanding complex algebraic expressions more easily.Key aspects of simplification include:

• Identifying like terms and combining them.
• Applying the laws of exponents to manage terms with both similar and different bases.
• Factorizing where necessary to simplify expressions.

In rationalizing the denominator of \(\frac{4}{\sqrt[3]{a^2}}\), we multiply by \(a^{1/3}\) both in the numerator and the denominator. This action neutralizes the radical, converting the denominator to a whole number. The expression \(\frac{4a^{1/3}}{a}\) then simplifies to \(4a^{-2/3}\) by subtracting the exponents of the terms. This final step offers a simplified, standard expression with no radicals in the denominator. Mastering these techniques in algebra offers you the tools to tackle a wide array of mathematical challenges.