Problem 80 - Precalculus: Functions and Graphs

Question

Simplify the expression, and rationalize the denominator when appropriate. $$\sqrt[3]{(2 r-s)^{3}}$$

Step-by-Step Solution

Answer

The expression simplifies to \( 2r - s \).

Step 1: Identify the Expression Type

The given expression is a cube root: \( \sqrt[3]{(2r-s)^3} \). Our task is to simplify it.

Step 2: Apply Cube Root Property

The cube root of a cube, \( \sqrt[3]{x^3} \), simplifies to \( x \). Therefore, we apply this to the given expression: \( \sqrt[3]{(2r-s)^3} = 2r-s \).

Step 3: Check for Additional Simplification

Since \( 2r-s \) is already simplified and there is no need to rationalize the denominator, our solution is complete.

Key Concepts

Cube Root

The cube root is a special type of root that is concerned with determining which number multiplies by itself three times to provide a given value. In mathematical terms, if we have a number expressed as the cube of another number, like \[x^3 = y\], the cube root of y, \( \sqrt[3]{y} \), returns the original number x. Understanding this concept is particularly useful when simplifying expressions of the form\( \sqrt[3]{a^3} \).

• The cube root essentially "undoes" the cubing operation.
• This is powerful in simplifying expressions since it allows you to bring complicated terms to their most straightforward form.

In the given example, the expression\( \sqrt[3]{(2r-s)^3} \)involves taking the cube root of a cubed term. Since the cube root function negates the cubic power, the expression simplifies neatly to \(2r-s\).

Understanding cube roots and their properties provides a strong foundation for handling radical expressions effectively.

Rationalizing the Denominator

Rationalizing the denominator is a technique used in simplifying expressions where the denominator involves a radical. While our exercise here doesn't involve this process because there wasn't a fractional expression needing rationalization, it is invaluable in various mathematical problems to ensure cleaner numerical expressions. Here's how it works:

• For a denominator involving a square root, you multiply the numerator and denominator by that root to eliminate the radical.
• Similarly, for cube roots and higher, you multiply by the appropriate power that achieves a non-radical denominator.

The process ensures that you have a rational number in the denominator, making it easier to understand and use in further mathematical operations. While removing radicals is not required in our solution of \(2r-s\), recognizing when and how to rationalize denominators sharpens your overall algebraic skills.

Exponentiation

Exponentiation involves raising a number or expression to a power, indicating how many times it should be multiplied by itself. With cube roots and related simplifications like \( \sqrt[3]{(2r-s)^3} \), understanding exponentiation becomes crucial.

• In our exercise, the base term \((2r-s)^3\) was initially raised to the third power.
• Applying the cube root reversed this operation, showing the tight relationship between roots and powers.
• This relationship implies that cube roots are the natural opposites of cubing.

The comprehension of exponentiation and its inverses, like cube roots, helps to navigate through seemingly complex algebraic structures effortlessly. The transformation from \((2r-s)^3\) to \(2r-s\) exemplifies the elegance of exponentiation when used alongside cube roots.

Expression Simplification

Expression simplification is the practice of reducing mathematical expressions to their simplest form, which helps in understanding and calculating with them efficiently. By applying basic rules, such as those involving cube roots and exponentiation, simplification turns complex ideas into accessible ones. In our problem,\( \sqrt[3]{(2r-s)^3} \)simplified directly to \(2r-s\).

Key aspects of expression simplification include:

• Recognizing when expressions can be reduced using algebraic identities or properties.
• Simplifying helps minimize computational errors and enhances logical reasoning.
• Checking if additional reduction is necessary, such as rationalizing denominators or further factoring.

In every mathematical task, a simplified expression holds clarity and correctness, paving the way for deeper, more complex exploration of algebra and calculus.