Problem 52 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 47-52: Rewrite the expression using rational exponents. $$ \sqrt[3]{r^{3}-s^{3}} $$

Step-by-Step Solution

Answer

\((r^3 - s^3)^{1/3}\)

Step 1: Understand the Radical Expression

The given expression is a cube root: \( \sqrt[3]{r^3 - s^3} \). In this form, cube roots can be expressed using rational exponents.

Step 2: Express the Radical as a Rational Exponent

To convert the cube root into an expression with a rational exponent, remember that \( \sqrt[3]{x} = x^{1/3} \). Apply this rule: \( \sqrt[3]{r^3 - s^3} = (r^3 - s^3)^{1/3} \).

Key Concepts

Cube Roots

Cube roots are an essential part of working with radical expressions. They help us find a number which, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2, because 2 multiplied by itself three times (2 \(\times\) 2 \(\times\) 2) equals 8. Cube roots are expressed using the radical symbol with a small 3 above, as in \( \sqrt[3]{8} \). You'll often encounter cube roots in various mathematical contexts.
To convert a cube root into a different form, specifically using rational exponents, you can remember this simple rule:

• The cube root of a number \( x \) is equivalent to raising \( x \) to the power of \( \frac{1}{3} \), i.e., \( \sqrt[3]{x} = x^{1/3} \).
Understanding this conversion is crucial when dealing with both simple and complex expressions that involve cube roots.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and higher-order roots. A radical expression usually contains a radical symbol \( \sqrt{} \), which indicates the root type. In exercises, you might need to simplify these expressions or rewrite them differently.
Consider the expression \( \sqrt[3]{r^3 - s^3} \). This expression uses a cube root radical to encompass the expression \( r^3 - s^3 \).

To rewrite radical expressions using rational exponents:

• The goal is to make calculations easier or prepare for further algebraic operations.
• Use the basic rule: \( \sqrt[n]{x} = x^{1/n} \) to convert the root into an exponent.
These transformations are not only important in basic algebra but also help in more advanced mathematical contexts. By rewriting radical expressions with rational exponents, you open the door to using exponent rules and simplifying further.

Expression Simplification

Expression simplification is a fundamental skill in mathematics, which involves making expressions easier to understand and work with. It's akin to tidying up a room - everything becomes clearer and more accessible.
In the context of the given problem, simplifying involves transforming \( \sqrt[3]{r^3 - s^3} \) into \( (r^3 - s^3)^{1/3} \) using rational exponents. This step is essential because:

• It makes further algebraic manipulation possible, enabling the use of exponent laws to simplify expressions further.
• It provides clarity, presenting the radical expression in a form that might be easier to handle in calculus or more advanced algebraic operations.
Always aim for a simpler expression - it can eliminate potential errors in subsequent steps and makes your mathematical work more elegant and efficient.